Question

in how many ways can 6 people be selected out of 13 people waiting in an interview queue, if a specified person is always included?

Answer 1

There are 792 ways to select 6 people from 13, with 1 specified person always included.

Step-by-step explanation:

In the situation where 6 people must be selected from a queue of 13 people, and a specific person must always be included, we start with a secured spot for that specified person. With that person already chosen, we are left with 5 people to select out of the remaining 12.

This is a problem of combinations, where order does not matter, and can be represented mathematically using the combination formula which is

C(n, k) = n! / (k!(n-k)!),

where 'n' is the total number of people to choose from (excluding the specified person now), 'k' is the number of people we still need to choose, '!', the factorial, represents the product of all positive integers up to the number.

The calculation will be C(12, 5) which represents the number of ways to select 5 people from 12. Performing the calculation, we get :

`{12 \choose 5} = (12!)/(5!(12-5)!)`

`{12 \choose 5} = (12 * 11 * 10 * 9 * 8)/(5 * 4 * 3 * 2 * 1)`

`{12 \choose 5} = 792`

Therefore, there are 792 different ways to select the remaining 5 people, and thus 792 different combinations in total with the specified person included.