The number of seven digits odd numbers, that can be formed using all the
seven digits 1, 2, 2, 2, 3, 3, 5 is ____________.

Solution

Maths sir · 3 steps

Fix the units digit as 1, 3 or 5, then arrange the other six with their repeats taken care of.

1. Odd means last digit. A number is odd exactly when its units digit is odd. From these digits, the units place can be 1, 3 or 5. Each choice leaves a different set of six digits, so count each case separately.

   Arrangements of n digits with repeats = n!/(p! q!…)

   Units digit	Remaining six digits	Arrangements
   1	2, 2, 2, 3, 3, 5	6!/(3! 2!) = 720/12 = 60
   3	1, 2, 2, 2, 3, 5	6!/3! = 720/6 = 120
   5	1, 2, 2, 2, 3, 3	6!/(3! 2!) = 720/12 = 60

2. Why 3 gives more. Putting one 3 in the units place leaves only a single 3 behind. Now the 2s are the only repeat, so you divide by 3! alone.
3. So. Total odd numbers = 60 + 120 + 60 = 240.

The common slip: 180. Dividing by 3! 2! in every case gives 60 + 60 + 60 = 180. Once a 3 sits in the units place, only one 3 remains, so that case is 6!/3! = 120.

The idea to keep. When digits repeat, fix the restricted place first, then recount the repeats in what is left. The repeat counts change from case to case, so never reuse one formula for every case.

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