Let S be the sample space of all five digit numbers. It p is the probability that a randomly selected number from S, is a multiple of 7 but not divisible by 5 , then 9 p is equal to :

Solution

Maths sir · 4 steps

Count multiples of 7, then take away the multiples of 35, since those are the ones divisible by 5.

1. Sample space. Five-digit numbers run from 10000 to 99999, so there are 99999 - 10000 + 1 = 90000 of them.
2. Multiples of 7. The first one is 7 × 1429 = 10003 and the last is 7 × 14285 = 99995. That gives 14285 - 1429 + 1 = 12857 multiples of 7.
3. Remove multiples of 35. A multiple of 7 that is also divisible by 5 is a multiple of 35. The first is 35 × 286 = 10010 and the last is 35 × 2857 = 99995, so there are 2857 - 286 + 1 = 2572 of them.

   9p = 9 × (12857 - 2572)/90000 = 10285/10000

4. So. There are 10285 favourable numbers, so 9p = 1.0285, which is option (c).

The idea to keep. To count multiples of k in a range, find the first and last multiple and use last minus first plus one. For "multiple of a but not of b", subtract the multiples of lcm(a, b).

Answer from the NTA final answer key. Solution by PrepWiser.