If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6 !) k, then k is equal to :

Solution

Maths sir · 4 steps

Count all arrangements, subtract the ones where C and S stick together, then divide by 6!.

1. Total arrangements. MATHEMATICS has 11 letters, with M, A and T each appearing twice. So all arrangements number 11!/(2! 2! 2!) = 11!/8.
2. C and S together. Glue C and S into one block. That leaves 10 units with the same three repeated pairs, giving 10!/8. The block can be CS or SC, so multiply by 2 to get (2 × 10!)/8.

   11!/8 - (2 × 10!)/8 = (10! (11 - 2))/8 = (9 × 10!)/8

3. Pull out 6!. Write 10! = 6! × 5040, since 10 × 9 × 8 × 7 = 5040. Then the count is 6! × (9 × 5040)/8 = 6! × 5670.
4. So. k = 5670, which is option (a).

The idea to keep. For "do not come together", count the total, glue the pair into one block, remember the block can flip both ways, and subtract. Taking out the common factorial early keeps the arithmetic small.

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