Question

In how many ways can a cricket-eleven be chosen out of 16 players such that a particular player is always chosen?

Answer 1

Hint: Here, there are 16 players out of which 11 players should be chosen such that a particular player is always chosen. So, we have to form a combination on the basis of the given condition. So, we need to make a combination of 10 players out of 15 players. This is because of the given condition in the question that one particular player should always be chosen. So, that is why we have subtracted 1. Then apply the combination formula for choosing 10 players out of 15.

Complete step by step solution:
Given, Total number of players = $16$
Total number of players is chosen = $11$
According to the given condition in the question we need a particular player is always chosen
This means one player is always the same in the team, so we have to choose the remaining team from the remaining members.
Thus, Total number of players we have now $= 16 – 1 = 15$
Total number of players is chosen $= 11 – 1 = 10$
[Combination Formula: Choosing $r$ items from total of $n$ item can be done in \[^n{C_r}\] ways]
Here, $n = 15$ and $r = 10$
Number of ways players will be chosen is $^{15}{C_{10}}$
$ = \dfrac{{15!}}{{10!5!}} = \dfrac{{15 \times 14 \times 13 \times 12 \times 11 \times 10!}}{{10! \times 5 \times 4 \times 3 \times 2 \times 1}} = 3003$

$\therefore$ Number of ways players will be chosen = 3003.

Note:
We have used the concept and formula of combination for solving the above question. Combination is a way of selecting items from a collection, such that (unlike permutation) the order of selection does not matter. Let’s see some real-life examples of combination – Selection of menu, food, clothes, subjects, the team are examples of combination while Arranging people, digits, numbers, alphabets, letters are examples of permutation.
In this type of question where a particular member is present in all combinations, exclude that member while calculating the number of ways.