Question

What is the probability that a random permutation of the letters in MATHEMATICS contains the word TEA (with those three letters appearing consecutively)?

Answer 1

The probability that a random permutation of the letters in MATHEMATICS contains the word TEA is 0.04.

Explanation:

First compute the number of permutations of the word MATHEMATICS.

There are a total of 11 letters.

And there are 2 Ms, 2 As and 2 Ts

Number of permutations of MATHEMATICS =
`(11!)/(2!*2!*2!) = 4989600`

Now, if the letters TEA occur together in the same order then the word may be of the form: (TEA)MHMATICS.

There are a total of 9 letters.

The letters TEA is treated as only one letter.

In this case there are 2 Ms.

Number of permutations of (TEA)MHMATICS =
`(9!)/(2!) = 181440`

The probability that the letters TEA occur together is:

`(Number\ of\ permutations\ of\ MATHEMATICS)/(Number\ of\ permutations\ of\ (TEA)MHMATICS) = (181440)/(4989600) \\=0.0363636\\\approx0.04`

Thus, the probability that a random permutation of the letters in MATHEMATICS contains the word TEA is 0.04.