Suppose you have a group of 10 people a How many ways are th

Suppose you have a group of 10 people.

(a) How many ways are there to choose a group of 4 people from these 10?

(b) How many ways are there to choose a group of 4 people from these 10 if two particular people (say, John and Dave) can not be on the committee together?

(c) How many ways are there to separate these 10 people into two groups, if no group can have less than 2 people? [Hint: when dividing into two groups, you only have to figure out who is in one group, since then the other one is determined.]

(d) How many ways are there to separate these 10 people into two groups of 5 each? [Hint: there is a subtle issue of overcounting here and in (c).]

Solution

Given that , there are group of 10 peoples.

(a) How many ways are there to choose a group of 4 people from these 10?

That means we have to choose people from 10.

Here we used term as combination.The formula for this is,

(n C x) = n! /(x! * (n-x)! ) (C is used for combination)

Here we have to calculate (10 C 4).

Then we use formula here ,

(10 C 4) = 10! / (4! * (10-4)! )

=10! / 4! * 6!

10! = 10*9*8*7*6*5*4*3*2*1 = 3628800

4! = 4*3*2*1 = 24

6! = 6*5*4*3*2*1 = 720

(10 C 4 ) = 3628800 /(24 * 720)

=210

(b) How many ways are there to choose a group of 4 people from these 10 if two particular people (say, John and Dave) can not be on the committee together?

Here also we have to select a group of 4 people from 10 and given that two peoples cannot be on the commette together.

Here (10 C 3) *( 7 C 1)

that is we have to choose first 3 peoples from 10 and then 1 people from (10-3=7) peoples.

(10 C 3) *( 7 C 1) = 120*7 = 840