The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 5 and 7 such that each digit is used at least once, is :

Step-by-step solution

A seven-digit number is to be formed using the 5 digits 1, 2, 3, 5, and 7 . Since each digit must be used at least once, the remaining 2 places can be filled by either repeating one digit two more times, or repeating two digits one more time each. This gives rise to two cases for the frequencies of the digits: Case 1: One digit appears 3 times and the other four digits appear 1 time each. The number of ways to choose the digit that appears 3 times is ⁵C₁ . The number of arrangements of these 7 digits is 7! 3! . Num