# COMBINATION PROBLEMS WITH SOLUTIONS

Combination Problems With Solutions :

Here we are going to see some practice questions base d on the concept combination.

## Combination Problems With Solutions

Problem 1 :

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 :

Number of white balls  =  2

Number of black balls  =  3

Number of red balls  =  4

Number of non black balls  =  2 + 4  =  6

 Number of black balls to be drawn Number of red balls to be drawn Total balls to be drawn 3C1 6C2 3 3C2 6C1 3 3C3 6C0 3

Number of ways

=  (3C 6C2) + (3C 6C1 +  (3C 6C0)

=  (⋅ 15) + (⋅ 6) + (1 ⋅ 1)

=  45 + 18 + 1

=  64

Problem 2 :

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

Solution :

There are 11 letters not all different.

They are AA, II, NN, E, X, M, T, O.

The following combinations are possible:

Case 1 :

Number of ways selecting 2 alike, 2 alike

=  3C2   =  3 ways

Case 2 :

Number of ways selecting 2 alike,2 different

=  3C  7C2 ==> 3 x 21  ==>  63 ways.

Case 3 :

Number of ways selecting all 4 different =  8C4

= 70 ways.

Total number of combinations = 3 + 63 + 70 = 136 ways.

Total number of permutations (1) to (3)

=  3  (4!/2!2!) + 63  (4!/2!) + 70  4!

=  18 + 756 + 1680

= 2454

Problem 3 :

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

Solution :

From the given question, we come to know that any three points are not collinear.

By selecting any three points out of 15 points, we draw a triangle.

Number of ways to draw a triangle  =  15C3

=  (15  14  13) / (3 ⋅ 2 ⋅ 1)

=  455

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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