How to count the number of subsets with property P

Question

Suppose, I want to create subset of cardinality $3$ from the set $\{1,2,3,4,5\}$. The total number of subset of cardinality $3$ is $\binom{n}{3}$. Out of these subsets I want to count the number of subsets starting with $1$. For example $123,124,125,134,135,145$ there are total $6$ subset of cardinality $3$ starting with $1$. Is there any simple equation to count this?

Answer 1

If you fix the first element, it's as if you can't choose it. So you're left with the problem of choosing 2 elements out of 4.

Answer 2

The number of ways to choose the remaining $2$ elements from the $4$-element set $\{2,3,4,5\}$ is ${4 \choose 2}=6$.