Question

Wish to select 6 persons from 8, but if person A is chosen then person B must be chosen. In how many ways the selection can be made?

Answer 1

Hint: Combination can be well-defined as the selection of the items where the order of the items does not matter and it gives the number of ways of the selection of the items and is expressed as ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ . In this question, there is a compulsion that if A is selected, then B also has to be selected, therefore in the cases which we are going to consider won’t have a case in which only A is selected.

Complete answer:To find: number of ways
As we want the number of ways, we should consider all the possibilities that may occur.
Therefore while selecting 6 persons from 8, with the specified condition in the question, there are 3 possibilities that can be considered. That are:
A and B are chosen
A and B are not chosen
Only B is chosen, A is not chosen
Now we have to look at the number of ways for each of the possibilities mentioned above.
A and B are chosen
Both A and B can be chosen in 6C4 ways, as from 6 persons, already 2 are chosen, so we are left with 4 persons to choose from.
Using formula, we get
${}^6{C_4} = \dfrac{{6!}}{{4!(6 - 4)!}} = \dfrac{{30}}{2}$
${}^6{C_4} = 15$ ---1
A and B are not chosen
Both A and B cannot be chosen in 6C6 ways, as from 6 persons, nobody is chosen, so we are left with 6 persons to choose from.
Using formula, we get
${}^6{C_6} = \dfrac{{6!}}{{6!(6 - 6)!}} = \dfrac{{6!}}{{6!}}$
${}^6{C_6} = 1$ ---2

Only B is chosen, A is not chosen
Only B can be chosen in 6C5 ways, as from 6 persons, only B is chosen, so we are left with 5 persons to choose from.
Using formula, we get
${}^6{C_5} = \dfrac{{6!}}{{5!(6 - 5)!}} = \dfrac{{6!}}{{5!}}$
${}^6{C_5} = 6$ ---3
Adding equations 1, 2 and 3, we get,
${}^6{C_4} + {}^6{C_5} + {}^6{C_6} = 6 + 15 + 1 = 22$
Therefore, the total number of ways for selecting are 22.

Note:
Permutations can be defined as the number of ways of the selection and the arrangement of the items in which order of the item is important and it is expressed as \[{}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\] Also remember the standard formulas and the difference between the two.