**Question 1. lf ^{n}C_{8} = ^{n}C_{2}, find ^{n}C_{2}.**

**Solution.**

We have, ^{n}C_{8} = ^{n}C_{2}

**Question 2. Determine n if**

**(i)**^{2n}C_{3}:^{n}C_{3}=12 : 1**(ii)**^{2n}C_{3}:^{n}C_{3}= 11 : 1

**Solution.**

**Question 3. How many chords can be drawn through 21 points on a circle?**

**Solution.**

A chord is formed by joining two points on a circle.

∴ Required number of chords = ^{2n}C_{2}

**Question 4. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?**

**Solution.**

3 boys can be selected from 5 boys in ^{5}C_{3} ways & 3 girls can be selected from 4 girls in ^{4}C_{3} ways.

∴ Required number of ways of team selection = ^{5}C_{3} x ^{4}C_{3} =

**Question 5. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.**

**Solution.**

No. of ways of selecting 3 red balls =^{6}C_{3}

No. of ways of selecting 3 white balls = ^{5}C_{3}

No. of ways of selecting 3 blue balls = ^{5}C_{3}

∴ Required no. of ways of selecting 9 balls

**Question 6. Determine the number of 5 cards combinations out of a deck of 52 cards if there is exactly one ace in each combination.**

**Solution.**

Total no. of cards = 52

No. of ace cards = 4

No. of non-ace cards = 48

∴ One ace card out of 4 can be selected in ^{4}C_{1} ways.

Remaining 4 cards out of 48 cards can be selected in ^{48}C_{4}ways.

∴ Required no. of ways of selecting 5 cards

**Question 7. In Kbw many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?**

**Solution.**

Total players = 17, No. of bowlers = 5,

No. of non-bowlers = 12

No. of ways of selecting 4 bowlers = ^{5}C_{4}

No. of ways of selecting 7 non-bowlers = ^{12}C_{7}

∴ Required no. of ways of selecting a cricket team

**Question 8. A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.**

**Solution.**

No. of ways of selecting 2 black balls = ^{5}C_{2}

No. of ways of selecting 3 red balls = ^{6}C_{3}

∴ Required no. of ways of selecting 2 black & 3 red balls = ^{5}C_{2} x ^{6}C_{3}

**Question 9. In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?**

**Solution.**

Total no. of courses = 9

No. of compulsory courses = 2

So, the student will choose 3 courses out of 7 courses [non-compulsory courses].

∴ Required no. of ways a student can choose a programme = ^{7}C_{3} =