In how many ways can a committee with eight members form a subcommittee with at least one person on it?

Question

is there a formula for this kind of questions?

Answer 1

Use:  nCr = n! / (r!(n-r)!)

 

8Choose1 + 8Choose2 + 8Choose3 + 8Choose4 + 8Choose5 + 8Choose6 + 8Choose7 + 8Choose8 

 

8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 255 ways

 

8 possibilities when choosing just 1; 28 possibilities when 2 are chosen; 56 possibilities when 3 are chosen;... 1 possibility when choosing 8.

 

Answer 2

"at least one person" means the subcommittee can have 1, 2, 3, 4, 5, 6, or 7 people on it (assuming you don't want a subcommittee with 8 people on it)

use the combination formula for each number of people on the subcommittee and then add the answers you get

the formula is n!/r!(n-r)! where n=the number of people (8) on the committee and r=the number of people you want on the subcommittee

n! means n*(n-1)*(n-2)*(n-3)*...*(2)*(1)

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

1 person subcommittee:

8!/1!(8-1)!=8/1=8 ways

2 person subcommittee:

8!/2!(8-2)!=(8*7)/2=28 ways

3 person subcommittee:

8!/3!(8-3)!=(8*7*6)/(3*2)=56 ways

4 person:

8!/4!(8-4)!=(8*7*6*5)/(4*3*2)=70 ways

5 person:56 ways

6 person:28 ways

7 person:8 ways

add: 8+28+56+70+56+28+8=254 ways