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

Answer:

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

Step-by-step 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 = $\frac{11!}{2!\times2!\times2!} = 4989600$

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 = $\frac{9!}{2!} = 181440$

The probability that the letters TEA occur together is:

$\frac{Number\ of\ permutations\ of\ MATHEMATICS}{Number\ of\ permutations\ of\ (TEA)MHMATICS} = \frac{181440}{4989600} \\=0.0363636\\\approx0.04$

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