COMBINATIONS WORD PROBLEMS EXAMPLES

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 :

Number of ways of selecting 7 players out of 14 players 

=  ¹⁴C₇ 

=  14! / (14 - 7)! 7! 

=  14! / 7! 7!

=  (14 ⋅ 13 ⋅ 12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8 ⋅ 7!)/7! 7!

=  (14 ⋅ 13 ⋅ 12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8) / 7!

=  (14 ⋅ 13 ⋅ 12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8) / (7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2)

=  3432

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 :

Before going to look into the solution of this problem, let us create a model. 

By comparing the above results, we may conclude the formula to find number of hand shakes

Number of hand shakes  =  n (n - 1)/2

Here we divide n (n - 1) by 2, because of avoiding repetition.

Number of persons in a party  =  15

Number  of hand shakes can be made  =  15 (15 - 1) / 2

  =  15 (14)/2

  =  15 (7)

  =  105

Example 3 :

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

Solution :

20 points lie on the circle. By joining any two points on the circle, we may draw a chord.

Number of chords can be drawn  =  ²⁰C₂

  =  20!/(20 - 2)! 2!

  =  20! / 18! 2!

  =  (20 ⋅ 19) / 2 

  =  190

Hence the required number of chords can be drawn is 190.

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 :

In the given question, we have a word "different set of five cars".

So we have to use the concept combination.

Number of ways of choosing 5 cars  =  ¹⁰⁰C₅

Hence the answer is ¹⁰⁰C_5.

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 :

Total number of ways  =  10 ⋅ 6 ⋅ 2

  =  120 ways

Hence the total number of ways is 120.

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 :

Total number of letters in the word MONDAY = 6

i) Out of 6 letters 4 letters are used at a time

number of words to be created = ⁶ P ₄

= 6!/(6 - 4)!

= 6! / 2!

= (6 x 5 x 4 x 3 x 2!)/2!

= 6 x 5 x 4 x 3

= 360

Vowels are A, O (so totally 2 letters)

ii) First letter should be vowel, the other letters may be anything vowel or consonants.

= ²P₁ x 5!

= 2 x 120

= 240

There are 240 possible words where all letters are used and the first letter is a vowel.

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 :

Total number of black balls = 5

Total number of red balls = 6

Number of ways = ⁵C₂ x ⁶C₃

= 5!/(5 - 2)!2! + 6!/(6 - 3)!3!

= 5!/3!2! + 6!/3!3!

= (5 x 4 x 3!/3!2!) + (6 x 5 x 4 x 3!/3!3!)

= (5 x 4/2!) + (6 x 5 x 4/3!)

= 5 x 2 + 5 x 4

= 10 + 20

= 30

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 :

Number of ways boys to be seated = 3!

Number of ways girls to be seated = 5!

Total number of ways = 3! x 5! x ⁶C₃

= 6 x 120 x 6!/3!3!

= 720 x (6 x 5 x 4 x 3!/3!3!)

= 720 x 5 x 4

= 14400

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 :

Total number of letters = 8

Number of vowels = 4

Number of consonants = 4

Total number of ways = (⁴C₃ x ⁴C₂) x 5!

= 4 x 6 x 120

= 2880

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.