# Problem of the Day

The letters in the word MATHEMATICIAN are written out in the following sequence: MAATTTHHHH…. so that the nth letter of the word is repeatedÂ n times. If a letter is selected at random from this sequence, what is the probability that it is an ‘A’?

A.
B.
C.
D.
E.

[latexpage]

C. $\frac{3}{13}$

See the Solution

### Solution

[latexpage]

MATHEMATICIAN has 3 A’s: one in the 2nd position, one in the 7th position, and one in the 12th position. That means in this sequence there will be $2+7+12=21$ A’s. The total number of letters in the sequence Â is $1+2+3+…+13$. This is an arithmetic series. The sum of an arithmetic series can be determined by taking the average of the first and last terms and multiplying this by the number of terms: $S=\frac{1+13}{2}\cdot13=7\cdot13=91$. So 21 out of 91 letters are A’s, so the probability of selecting an A is $\frac{21}{91}=\frac{3}{13}$.

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