The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _________.

Solution

Maths sir · 3 steps

Split 6 into parts of at least 2, choose letters for each part, then arrange with repeats.

1. Possible patterns. MATHS has 5 different letters. Each letter that is used must appear at least twice, so the counts must add to 6 with every part at least 2. The only patterns are 6, 4+2, 3+3 and 2+2+2.
2. Count each pattern. For each pattern, choose which letters fill the parts, then arrange them with 6!/(product of factorials of counts). In 4+2 the two parts are different, so the choice is ordered. In 3+3 and 2+2+2 the parts are equal, so the choice is a combination.

   Pattern	Choose letters	Arrangements	Words
   6	5	6!/6! = 1	5
   4+2	5 × 4 = 20	6!/(4! 2!) = 15	300
   3+3	C(5, 2) = 10	6!/(3! 3!) = 20	200
   2+2+2	C(5, 3) = 10	6!/(2! 2! 2!) = 90	900

3. So. The total is 5 + 300 + 200 + 900 = 1405.

The common slip: 1255. Using C(5, 2) = 10 for the 4+2 case misses that the letter used 4 times differs from the one used twice. That gives 150 there and 1255 in total.

The idea to keep. For conditions on how often letters repeat, first list the count patterns. Choose letters for unequal counts in order, for equal counts as a combination, then divide n! by the repeat factorials.

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