Evaluate $\frac{n!}{r!(n-r)!}$ , when $n = 5, r = 2$.

Step 1: Substitute the values of n and r

The problem asks us to evaluate the given expression by substituting the values $n=5$ and $r=2$. We replace $n$ with 5 and $r$ with 2 in the expression.

Step 2: Simplify the expression

Next, we simplify the term inside the parenthesis in the denominator. $5-2$ equals $3$, so the expression becomes $\frac{5!}{2!3!}$.

Step 3: Expand the factorials

Now, we expand each factorial term. Recall that $n!$ (n factorial) is the product of all positive integers less than or equal to $n$. So, $5! = 5 \times 4 \times 3 \times 2 \times 1$, $2! = 2 \times 1$, and $3! = 3 \times 2 \times 1$.

Step 4: Calculate the products

We calculate the product for each factorial. $5 \times 4 \times 3 \times 2 \times 1 = 120$. For the denominator, $2 \times 1 = 2$ and $3 \times 2 \times 1 = 6$.

Step 5: Perform the final division

Finally, we multiply the terms in the denominator: $2 \times 6 = 12$. Then, we divide the numerator by the denominator: $120 \div 12 = 10$.

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