Question 5 and 6, Exercise 6.3

Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

In how many ways can $11$ players be chosen out of $16$ if there is no restriction.

Solution.

$11$ players out of $16$ may be chosen in ${ }^{16} C_{11}$ ways.
i.e. $4368$ ways are there to choose $11$ players.

In how many ways can $11$ players be chosen out of $16$ if a particular player is always chosen.

Solution.

If one player is always chosen then we can only choose $10$ players out of $15$ so-this time number of ways are

$${ }^{15} C_{10}=3003$$

Out of $5$ men and $3$ women, a committee of $3$ is to be formed. In how many ways can it be formed if t least one men is selected?

Solution.

Case I: If one man is chosen there will be two women in committee.
$1$ man may be selected ${ }^{5} C_{1}$ ways
and $2$ women may be selected in ${ }^{3} C_{2}$ ways.
Total possible ways $={ }^{5} C_{1} \times{ }^{3} C_{2}=5 \times 3=15$

Case II: If there are $2$ men and one woman in committee.

Total possibilities $={ }^{5} C_{2} \times{ }^{3} C_{1}=10 \times 3=30$

Case III: If there are $3$ men and no woman in committee.

Total possible ways $={ }^{5} C_{3}=10$
Hence a committee of $3$ having at least one man may be selected in $15+30+10=55$ ways

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