Probality and Statistics

Question

A driver comes to pick up members of your family for a family reunion. The van holds 7 people, not including the driver. Assuming that none of the babies will be riding in the van, how many different ways can 7 people be chosen to ride in the van? (15 famiy members) No babies

Please help with formula, steps and solution. Thank you.

Answer 1

Use the formula _nC_r, where n=population to choose from and r=subset of n that is chosen.

Here n=15 and r=7. Note that C says "choose a sample of size r from population of size n".

The formula _nC_r, in our case ₁₅C₇ = n!/[r! * (n-r)!].

The symbol "!" means "factorial", defined as n*(n-1)*(n-2)* ... * 1. 0! is defined as 1

n! = 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1

r! = 7*6*5*4*3*2*1

(n-r)! = (15-7)! = 8! = 8*7*6*5*4*3*2*1

Notice that n! includes all the terms in r! as well as (n-r)!

Factor out the terms for (n-r)!

This leaves us with 15*14*13*12*11*10*9/7*6*5*4*3*2*1

Simplify as much as possible.

15 = 5*3, 14 = 7*2.. Factor these out to get

13*12*11*10*9/6*4*1

Note that 6*4 = 2*12, so 12 can be factored out to give

13*11*10*9/2*1

Finally, 2 can be factored out to give the final number of ways 7 people out of 15 can be seated in the van as

13*11*5*9/1 = 6435