PRACTICE QUESTIONS ON COMBINATION

Question 1 :

If ⁿC₁₂ =ⁿC₉ find ²¹C_n.     Solution

Question 2 :

If ¹⁵C_2r−1 = ¹⁵C_2r+4, find r.    Solution

Question 3 :

If ⁿP_r = 720, and ⁿC_r = 120, find n, r.   Solution 

Question 4 :

Prove that ¹⁵C₃ + 2 × ¹⁵C₄ + ¹⁵C₅ = ¹⁷C₅.   Solution

Question 5 :

Prove that ³⁵C₅ + ∑ ⁴_r=0 ^(39−r) C₄  =  ⁴⁰C₅.  Solution 

Question 6 :

If ⁿ C ₁₂ = ⁿ C ₈, then n is equal to 

a) 20   b)  12   c) 6    d)  30

Solution

Question 7 :

Find r, If ¹⁵ C _r : ¹⁵ C _{r - 1} = 11 : 5

Solution

Question 8 :

If ⁿP_r = 336, ⁿC_r = 56. Find n and r and hence find ^n–1C_r–1.

Solution

Question 9 :

Find n if ²ⁿC₃ : ⁿC₃ = 11 : 1

Solution

Question 10 :

Show that (n + 2) n! = n! + (n + 1)!

Solution

Question 1 :

If ⁽ⁿ⁺¹⁾C₈ :⁽ⁿ⁻³⁾ P₄ = 57 : 16, find the value of n.  Solution

Question 2 :

Prove that ²ⁿC_n = [2n × 1 × 3 × ·· · (2n − 1)] / n!.  Solution

Question 3 :

Prove that if 1 ≤ r ≤ n then n × ⁽ⁿ⁻¹⁾ C_r−1 = (n − r + 1) ⁿC_r−1.

Solution

Question 4 :

Kabaddi coach has 14 players ready to play. How many different teams of 7 players could the coach put on the court?

Solution

Question 4 :

There are 10 points in a plane, out of which 4 points are collinear. The number of triangles formed with vertices at these points is

(a) 20        (b) 120       (c) 116      (d) none of these

Solution

Question 5 :

10 students are participating in a competition. In how many different ways can the first prize be won? (There are 3 prizes)

(a) 720     (b) 60        (c) 30       (d) 120

Solution

Question 6 :

Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to

(a) 60     (b) 120      (c) 7200       (d) 720

Solution

Question 7 :

All the letters of the word ‘EAMCOT’ are arranged in different possible ways. The number of such arrangements in which no two vowels are adjacent to each other is

(a) 360      (b) 144      (c) 72     (d) 54

Solution

Question 8 :

The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is

(a) 105     (b) 15    (c) 175     (d) 185

Solution

Example 1 :

Kabaddi coach has 14 players ready to play. How many different teams of 7 players could the coach put on the court?

Solution

Example 2 :

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

Example 3 :

How many chords can be drawn through 20 points on a circle?

Solution

Example 4 :

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

Example 5 :

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

Example 6 :

How many words, with or without meaning can be made from the letters of the word MONDAY. Assuming that no. letter is repeated, it

(i) 4 letters are used at a time

(ii) All letters are used but first letter is a vowel?

Solution

Example 7 :

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

Example 8 :

In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together?

Solution

Example 9 :

How many words, with or without meaning, each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE.

Solution

Problem 1 :

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

Problem 2 :

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?

Solution

Problem 3 :

Suppose that 7 people enter a swim meet. Assuming that there are no ties, in how many ways could the gold, silver, and bronze medals be awarded?

Solution

Problem 4 :

How many different committees of 3 people can be chosen to work on a special project from a group of 9 people?

Solution

Problem 5 :

A coach must choose how to line up his five starters from a team of 12 players. How many different ways can the coach choose the starters?

Solution

Problem 6 :

John bought a machine to make fresh juice. He has five different fruits: strawberries, oranges, apples, pineapples, and lemons. If he only uses two fruits, how many different juice drinks can John make?

Solution

Problem 7 :

How many different four-letter passwords can be created for a software access if no letter can be used more than once?

Solution

Problem 8 :

.How many different ways you can elect a Chairman and Co-Chairman of a committee if you have 10 people to choose from.

Solution

Problem 9 :

There are 25 people who work in an office together. Five of these people are selected to go together to the same conference in Orlando, Florida. How many ways can they choose this team of five people to go to the conference?

Solution

Problem 10 :

There are 25 people who work in an office together. Five of these people are selected to attend five different conferences. The first person selected will go to a conference in Hawaii, the second will go to New York, the third will go to San Diego, the fourth will go to Atlanta, and the fifth will go to Nashville. How many such selections are possible?

Solution

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