Question

A four-digit number is formed by using only the digits 1, 2 and 3 such that both 2 and 3 appear at least once. The number of all such four-digit numbers is

Answer 1

Correct Answer: 50

For the four-digit numbers using only the digits 1, 2, and 3 where both 2 and 3 appear at least once, we can break down the problem into cases:

Case 1: Two occurrences of 1, one of 2, and one of 3. The number of ways to arrange two 1s, one 2, and one 3 = 4!/2! = 12 ways. (Divide by 2! because there are two 1s that are identical.) 

Case 2: One occurrence of 1, two of 2, and one of 3. The number of ways to arrange one 1, two 2s, and one 3 = 4!/2! = 12 ways. 

Case 3: One occurrence of 1, one of 2, and two of 3. The number of ways to arrange one 1, one 2, and two 3s = 4!/2! = 12 ways. 

Summing up the number of ways from all the cases = 12 + 12 + 12 = 36. 

So, there are 36 such four-digit numbers