The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ___________.

Solution

Maths sir · 4 steps

Glue all six vowels into one block, arrange the blocks, then arrange the vowels inside the block.

1. Sort the letters. ASSASSINATION has 13 letters. The vowels are A, A, A, I, I, O (6 of them). The consonants are S, S, S, S, N, N, T (7 of them).
2. Arrange the units. Treat the vowel block as one unit, so there are 7 + 1 = 8 units. The S repeats 4 times and the N twice, so the units can be arranged in 8!/(4! 2!) = 40320/48 = 840 ways.
3. Arrange inside block. Inside the block, A repeats 3 times and I twice, so the vowels can be arranged in 6!/(3! 2!) = 720/12 = 60 ways.

   8!/(4! 2!) × 6!/(3! 2!) = 840 × 60

4. So. The number of words is 840 × 60 = 50400.

The common slip: 840. Stopping after arranging the 8 units gives 840. The 6 vowels can still be rearranged among themselves in 60 ways.

The idea to keep. For 'together' questions, make the group one unit and count the outside arrangements. Then multiply by the arrangements inside the group, dividing by repeats both times.

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