The total number of words (with or without meaning) that can be formed out of the letters of the word ' DISTRIBUTION ' taken four at a time, is equal to __________.

Solution

Maths sir · 3 steps

Repeated letters break the count, so split by how many letters repeat and count each case on its own.

1. Sort the letters. DISTRIBUTION has 12 letters but only 9 different ones. I appears 3 times, T appears 2 times, and D, S, R, B, U, O, N appear once each.
2. Split into cases. A 4-letter word can use 4 different letters, one pair and two singles, two pairs, or three alike plus one. For each case, choose the letters, then arrange them, dividing by the factorial of each repeat.

   Pattern	Choose the letters	Arrangements	Words
   All 4 different	C(9, 4) = 126	4! = 24	3024
   2 alike, 2 different	pair from {I, T}: 2, others: C(8, 2) = 28	4!/2! = 12	2 × 28 × 12 = 672
   2 alike, 2 alike	only IITT: 1	4!/(2! 2!) = 6	6
   3 alike, 1 different	III with one of 8 others: 8	4!/3! = 4	32

3. So. Total = 3024 + 672 + 6 + 32 = 3734.

The common slip: 3024. Treating the 9 distinct letters as the only choices and computing ⁹P₄ gives 3024. That misses every word that uses a repeated I or T.

The idea to keep. When a word has repeated letters and you pick only some of them, list every repeat pattern (all different, one pair, two pairs, three alike) and add the cases. Divide by the factorial of each repeat when arranging.

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