Maths is good for you

If a permutation of all 26 letters of the english alphabet is chosen at random, what is the probability that the permutation contain the letters MATHS in the same order (A doesn't come before M , T doesn't come before A , H doesn't come before T , S doesn't come before H). The probability can be expressed as m n \frac mn where m m and n n are positive coprime integers. Find the value of m + n m+n .

Details and Assumptions

For example: "MATHSBCDEFGIJKLNOPQRUVWXYZ" is a valid sequence and so is "MPAQTRHSBCDEFGIJKLNOUVWXYZ", as order of letters of MATHS is maintained


The answer is 121.

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1 solution

There will be the same number of permutations of all 26 26 letters for each of the 5 ! = 120 5! = 120 possible permutations of the letters M , A , T , H , S , M,A,T,H,S, so the probability that the letters M , A , T , H , S M,A,T,H,S appear in that precise order (as described in the question) in a randomly chosen permutation of the 26 26 letters of the English alphabet is 1 120 . \dfrac{1}{120}. Thus m + n = 1 + 120 = 121 . m + n = 1 + 120 = \boxed{121}.

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