Question

How many sets of five numbers from 1 to 15 can you make in which exactly two of the numbers are divisible by 3?

Answer 1

Given

The numbers are from 1 to 15 ; We have to make sets of five numbers in that exactly two of the numbers are divisible by 3.

Number that divisible by 3 in the range of 1 to 15 = 3,6,9,12,15 ( Total 5 numbers)

Now out of these 5 numbers, we have to select exactly 2 numbers, Total number of way to select 2 numbers from 5 numbers = 5C2 =  10

Now out of 10 numbers(1,2,4,5,7,8,10,11,13,14), we have to select 3 numbers.

Total number of way to select 3 numbers from 10 numbers = 10C3 = 120

Total Number of set of five numbers from 1 to 15 in which exactly two of the numbers are divisible by 3  = 5C2*10C3*5! ( I multiply by 5! because we can rearrange that selected 5 numbers)

Total Number of set = 5C2*10C3*5! = 10*120*120 = 144000

Total Number of set = 5C2*10C3 = 10*120 = 12000 ( Without arranging the numbers)