How many different committees can be formed from 7 people if each committee must consist of at least 3 people?

Answer 1

7C3 = 7!/4!3! = 7x6x5/3x2 = 35 different ways to form a committee of 3 from 7 people

7C3 means combinations of 7 taken 3 at a time

3 people forming committees of 2

has 3C2 = 3!/2!1! = 3 ways

call them persons A, B and C

committees are

AB

AC

or

BC

this assumes order doesn't matter, so that AB is the same as BA, AC=CA and BC=CB

if order matters there are more ways to form the committees, using permutations

some fairly inexpensive hand calculators will do the calulations for you.

they have a key nCr and nPr, for combinations and calculations

plug in 7 then hit nCr then hit 3 to get 35

plug in 7 then hit nPr, then hit 3 to get 210 way to form a committee of 3 out of 7, if the order matters, where ABC is not the same committee as BCA or CBA

BUT the question didn't say a committee of 3, it said a committee of at least 3

so the total ways are 7C3 + 7C4 + 7C5 + 7C6 + 7C7

= 35 + 35 + 21 + 7 + 1 = 99 ways to form a committee of at least 3

UNLESS order matters, then far more

do the same calculations but use nPr

total ways = 7P3 + 7P4 + 7P5 + 7P6 + 7P7

use a calculator and save some time