**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

**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