Example 1: Combination

In how many ways can a committee of 4 men and 2 women be formed from a group of 10 men and 12 women?

ANSWER

Here, we are choosing 4 from 10 and also 2 from 12.

So we have:

10C4 * 12C2

The formula for combinations nCr is:

n! / r! (n-r)!

10C4 = 10! / 4!(10-4)! =   (10*9*8*7*6*5*4*3*2*1) / (4*3*2*1) (6*5*4*3*2*1)

= (10*9*8*7) / (4*3*2*1)

= 10 * 3 * 7 = 210

Using Excel:

=COMBIN(10,4)

result is 210

12C2 = 12! / 2! (12-2)! = (12*11*10*9*8*7*6*5*4*3*2*1) / (2*1) (10*9*8*7*6*5*4*3*2*1)

= (12*11) / (2 * 1)

= 6*11

= 66

Using Excel

=COMBIN(12,2)

result is 66

Finally,

10C4 * 12C2

= 210 * 66 = 13860

 

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Example 2: Permutation

A teacher and 14 students are to be seated along a bench in the bleachers at a basketball game. In how many ways can this be done if the teacher must be seated at the left end only?

ANSWER

In this case, we have 15 people (14 students and a teacher). But, we are told that the teacher must sit on the left end.

So the answer here is

1P1 * 14P14

In other words, the number of ways to “permute” or move around 1 person is just 1.

The number of way to move around the 14 students is 14P14

Formula for Permutation

nPr = n!/(n – r)!

Therefore, 1P1 = 1!/(1 – 1)! = 1!/0! = 1

Note that 0! = 1

Next, 14P14 = 14!/(14 – 14)! = 14!/1 = 14!

14! = 14*13*12*11*10*…*1

In Excel you can use

=PERMUT(14,14) = 87178291200

or you can do this by hand.

The final answer is

87178291200