[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ 1 & a & b \\ 2 & 1 & a \end{bmatrix} where a, b, and c are integers between 3 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
Let M = \begin{bmatrix} a & b & c \\ b & a & b \\ c & b & a \end{bmatrix}, where a, b, and c are integers from 1 to 10 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ b & a & b & c \\ c & b & a & b \\ d & c & b & a \end{bmatrix}, where a, b, c, and d are integers from 2 to 5 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [3, 9], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ b & a & b \\ c & b & a \end{bmatrix} where a, b, and c are integers between 2 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 2?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c & d \\ b & a & b & c \\ c & b & a & b \\ d & c & b & a \end{bmatrix} where a, b, c, and d are integers between 1 and 6, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [2, 10], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} where a, b, and c are integers between 1 and 9, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ 1 & a & b \\ 2 & 1 & a \end{bmatrix} where a, b, and c are integers between 1 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [1, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c & d \\ b & a & 1 & 2 \\ c & 1 & b & 3 \\ d & 2 & 3 & a \end{bmatrix} where a, b, c, and d are integers between 2 and 6, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix}, where a, b, and c are integers from 1 to 5 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [1, 5], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} where a, b, and c are integers between 2 and 5, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c \\ b & a & b \\ c & b & a \end{bmatrix}, where a, b, and c are integers from 1 to 8 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [1, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ 0 & b & a & c \\ 0 & 0 & c & b \\ 0 & 0 & 0 & d \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [2, 6], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ 1 & a & b \\ 2 & 1 & a \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [1, 6], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ 0 & b & a & c \\ 0 & 0 & c & b \\ 0 & 0 & 0 & d \end{bmatrix}, where a, b, c, and d are integers from 3 to 6 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [3, 7], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [3, 9], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix} where a, b, and c are integers between 2 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix}, where a, b, and c are integers from 3 to 9 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ b & a & 1 & 2 \\ c & 1 & b & 3 \\ d & 2 & 3 & a \end{bmatrix}, where a, b, c, and d are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [3, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} where a, b, and c are integers between 3 and 8, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [1, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ b & a & 1 & 2 \\ c & 1 & b & 3 \\ d & 2 & 3 & a \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c & d \\ b & a & 1 & 2 \\ c & 1 & b & 3 \\ d & 2 & 3 & a \end{bmatrix} where a, b, c, and d are integers between 3 and 5, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix} where a, b, and c are integers between 3 and 9, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [2, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ b & a & 1 & 2 \\ c & 1 & b & 3 \\ d & 2 & 3 & a \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ 1 & a & b & c \\ 2 & 1 & a & b \\ 3 & 2 & 1 & a \end{bmatrix}, where a, b, c, and d are integers from 3 to 5 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c \\ 1 & a & b \\ 2 & 1 & a \end{bmatrix}, where a, b, and c are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [1, 7], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ b & a & b \\ c & b & a \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [2, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ b & a & 1 & 2 \\ c & 1 & b & 3 \\ d & 2 & 3 & a \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} where a, b, and c are integers between 3 and 9, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [3, 7], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ 0 & b & a & c \\ 0 & 0 & c & b \\ 0 & 0 & 0 & d \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [1, 8], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ b & a & b \\ c & b & a \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ b & a & b & c \\ c & b & a & b \\ d & c & b & a \end{bmatrix}, where a, b, c, and d are integers from 1 to 5 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ 0 & b & a & c \\ 0 & 0 & c & b \\ 0 & 0 & 0 & d \end{bmatrix}, where a, b, c, and d are integers from 1 to 7 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [2, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ 1 & a & b & c \\ 2 & 1 & a & b \\ 3 & 2 & 1 & a \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ 1 & a & b & c \\ 2 & 1 & a & b \\ 3 & 2 & 1 & a \end{bmatrix}, where a, b, c, and d are integers from 1 to 6 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c & d \\ b & a & b & c \\ c & b & a & b \\ d & c & b & a \end{bmatrix} where a, b, c, and d are integers between 2 and 7, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix}, where a, b, and c are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c \\ b & a & 1 \\ c & 1 & b \end{bmatrix}, where a, b, and c are integers from 1 to 9 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [2, 7], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, c, and d \in [3, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \begin{bmatrix} a & b & c & d \\ b & a & b & c \\ c & b & a & b \\ d & c & b & a \end{bmatrix} has rank 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
For integers a, b, and c \in [3, 8], count the number of distinct ordered triples (a, b, c) such that the matrix \begin{bmatrix} a & b & c \\ 0 & b & a \\ 0 & 0 & c \end{bmatrix} has rank 3.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & b & c & d \\ 1 & a & b & c \\ 2 & 1 & a & b \\ 3 & 2 & 1 & a \end{bmatrix} where a, b, c, and d are integers between 2 and 5, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let M = \begin{bmatrix} a & b & c & d \\ 0 & b & a & c \\ 0 & 0 & c & b \\ 0 & 0 & 0 & d \end{bmatrix}, where a, b, c, and d are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider the matrix M = \begin{bmatrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{bmatrix} where a, b, c, and d are integers between 3 and 5, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?

Present the answer in LaTex format: \boxed{Your answer}