Permutations and Combinations · Question 34

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

1. 5720
2. 5719
3. 4608
4. 4607

Answer

(4) 4607

Solution

Digits may repeat. A 5 digit number greater than 50000 must begin with 5, 6 or 7.

(First digit + last digit) ≤ 8:

First digit 5 ⇒ last digit 0, 1, 2, 3 ⇒ 4 ways

First digit 6 ⇒ last digit 0, 1, 2 ⇒ 3 ways

First digit 7 ⇒ last digit 0, 1 ⇒ 2 ways

(first, last) pairs = 4 + 3 + 2 = 9

Each of the three middle places can be filled in 8 ways 83 = 512

Total = 9 512 = 4608

This count includes 50000, which is not greater than 50000.

Required number = 4608 - 1 = 4607

Answer: (4)