Question

read the combination value directly from pascals triangle.
$_{5}c_{3}$
$_{5}c_{3}=square$

Explanation:

Step1: Recall Pascal's triangle rule

Pascal's triangle rows are numbered starting from 0. The $n$-th row gives the binomial coefficients $\binom{n}{k}$ for $k = 0,1,\cdots,n$.

Step2: Identify row and position

For $\binom{5}{3}$, we look at the 5 - th row (counting from 0) of Pascal's triangle. The elements of the $n$-th row of Pascal's triangle are given by $\binom{n}{k}=\frac{n!}{k!(n - k)!}$. In Pascal's triangle, the 5 - th row is 1, 5, 10, 10, 5, 1. The positions in the row are numbered from 0. For $\binom{5}{3}$, we take the 4 - th element (since we start counting positions from 0) of the 5 - th row.

Answer:

10