Find number of occurrences of $n$ in a combination

Question

list $= \{1, 2, 3, 4, 5\}$;

The above list has $5$ numbers. The number of combinations when we group them into $3$ is $10$.

The list if combinations are

$\{123, 124, 125, 134, 135, 145, 234, 235, 245, 345\}$

I need to calculate the number of occurrences of an item in the list. When $n = 5$, the number of occurrences of $n$ in the list is $6$.

Is there any formula to find the number of occurrences?

Answer 1

You are taking $3$ numbers out of $5$ numbers. According to you, the place of $5$ must be fixed. So, there are two places remaining with $4$ possible numbers .

(I) In how many ways can you fill up the two places?

Answer 2

If you are choosing combinations of k numbers from n,
each chosen number will have the same probability of occurrence, $\frac{k}{n}$

Thus the number of occurrences of any of them will be $\binom{n}{k}\cdot\frac{k}{n}$

For your example, $\binom53\cdot\frac35 =6$

Answer 3

If the list has $n$ numbers then there are $\binom{n}{k}$ combinations of $k$ numbers.

So in total $\binom{n}{k}k$ numbers occur in the combinations.

Then by symmetry each of $n$ numbers will occur: $$\frac1n\binom{n}{k}k=\binom{n-1}{k-1}$$times.