**Practice Questions on Combination :**

In this section, we will learn, how to solve problems on combinations using the practice questions given below.

1. If ^{n}C_{12} =^{n}C_{9} find ^{21}C_{n}. Solution

2. If ^{15}C_{2r−1} = ^{15}C_{2r+4}, find r. Solution

3. If ^{n}P_{r} = 720, and ^{n}C_{r} = 120, find n, r. Solution

4. Prove that ^{15}C_{3} + 2 × ^{15}C_{4} + ^{15}C_{5} = ^{17}C_{5}. Solution

5. Prove that ^{35}C_{5} + ∑ ^{4}_{r=0 }^{(39−r)} C_{4 }= ^{40}C_{5}. Solution

6. If ^{(n+1)}C_{8} :^{(n−3)} P_{4} = 57 : 16, find the value of n. Solution

7. Prove that ^{2n}C_{n} = [2n × 1 × 3 × ·· · (2n − 1)] / n!. Solution

8. Prove that if 1 ≤ r ≤ n then n × ^{(n−1)} C_{r−1} = (n − r + 1) ^{n}C_{r−1}.

9. (i) A Kabaddi coach has 14 players ready to play. How many different teams of 7 players could the coach put on the court? Solution

(ii) There are 15 persons in a party and if each 2 of them shakes hands with each other, how many handshakes happen in the party? Solution

(iii) How many chords can be drawn through 20 points on a circle? Solution

(iv) In a parking lot one hundred , one year old cars, are parked. Out of them five are to be chosen at random for to check its pollution devices. How many different set of five cars can be chosen? Solution

(v) How many ways can a team of 3 boys,2 girls and 1 transgender be selected from 5 boys, 4 girls and 2 transgenders? Solution

10. Find the total number of subsets of a set with

[Hint: ^{n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + · · · + ^{n}C_{n} = 2n]

(i) 4 elements (ii) 5 elements (iii) n elements Solution

11. A trust has 25 members.

(i) How many ways 3 officers can be selected? Solution

(ii) In how many ways can a President, Vice President and a Secretary be selected? Solution

12. How many ways a committee of six persons from 10 persons can be chosen along with a chair person and a secretary? Solution

13. How many different selections of 5 books can be made from 12 different books if,

(i) Two particular books are always selected? Solution

(ii) Two particular books are never selected? Solution

14. There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees

(i) a particular teacher is included? Solution

(ii) a particular student is excluded? Solution

15. In an examination a student has to answer 5 questions, out of 9 questions in which 2 are compulsory. In how many ways a student can answer the questions? Solution

16. Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination. Solution

17. Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee. Solution

18. A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of

(i) exactly 3 women?

(ii) at least 3 women?

(iii) at most 3 women? Solution

19. 7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife also has 7 relatives; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relative and 3 of the wife’ s relatives?

20. A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw? Solution

21. Find the number of strings of 4 letters that can be formed with the letters of the word EXAMINATION?. Solution

22. How many triangles can be formed by joining 15 points on the plane, in which no line joining any three points?

23. How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line? Solution

24. There are 11 points in a plane. No three of these lies in the same straight line except 4 points,which are collinear. Find,

(i) the number of straight lines that can be obtained from the pairs of these points?

(ii) the number of triangles that can be formed for which the points are their vertices? Solution

25. A polygon has 90 diagonals. Find the number of its sides? Solution

After having gone through the stuff given above, we hope that the students would have understood, how to solve problems on combinations.

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