The probability that a randomly chosen 5-digit number is made from exactly two digits is :

Solution

Maths sir · 4 steps

Split the count by whether 0 is one of the two digits, because 0 cannot lead a 5-digit number.

1. Sample space. 5-digit numbers run from 10000 to 99999, so there are 90000 = 9 × 10⁴ of them.
2. Both digits nonzero. Choose the two digits in C(9, 2) = 36 ways. Each of the 5 places takes either digit, giving 2⁵ = 32 strings. Remove the 2 strings that use only one digit, which leaves 30. That gives 36 × 30 = 1080 numbers.
3. One digit is zero. Choose the nonzero digit d in 9 ways. The first place must be d. The other 4 places take 0 or d, which is 2⁴ = 16 strings. Remove dddd, which has no zero, and 15 are left. That gives 9 × 15 = 135 numbers.

   P = (1080 + 135)/90000 = 1215/90000 = 135/10⁴

4. So. The probability is 135/10⁴, which is option (d).

The idea to keep. With digit-count questions, always treat 0 as a separate case. It cannot sit in the first place, so the cases with 0 and without 0 never count the same way.

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