The letters in the word MATHEMATICIAN are written out in the following

Question

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

A.  \frac{3}{91}

B.  \frac{23}{91}

C.  \frac{3}{13}

D.  \frac{5}{13}

E.  \frac{31}{91}

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rajatchopra1994 · Answer

Explanation:

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 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={(1+13)*13}/2
=7*13 = 91.
So 21 out of 91 letters are A’s, so the probability of selecting an A is 21/91 = 3/13.
IMO-C

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wilsonprj · Answer

Step 1: Total lenght of the sequence

total letters = 13*14/2 = 91

Word:
M A T H E M A T I C I A N
Positions:
1 2 3 4 5 6 7 8 9 10 11 12 13

“A” appears at positions: 2, 7, 12

Each position contributes its index number of repetitions:

Position 2 → 2 times

Position 7 → 7 times

Position 12 → 12 times

Total A’s = 2 + 7 + 12 = 21

Therefore: 21/91 = 3/13

egmat · Answer

Let me walk you through this step by step.

Step 1:  Write out the word MATHEMATICIAN letter by letter with positions.

M-A-T-H-E-M-A-T-I-C-I-A-N
That's  13  letters total.

Position  1 : M
Position  2 : A
Position  3 : T
Position  4 : H
Position  5 : E
Position  6 : M
Position  7 : A
Position  8 : T
Position  9 : I
Position  10 : C
Position  11 : I
Position  12 : A
Position  13 : N

Step 2:  Build the sequence. The rule says the nth letter gets repeated n times. So:
- Position  1  → M written  1  time
- Position  2  → A written  2  times
- Position  3  → T written  3  times
- ...and so on up to position  13 .

The sequence looks like: M, AA, TTT, HHHH, EEEEE, MMMMMM, AAAAAAA, ...

Step 3:  Find the total number of letters in the sequence.
 Total = 1 + 2 + 3 + ... + 13 = (13 × 14) / 2  =  91

Step 4:  Find how many A's are in the sequence.
A appears at positions  2 ,  7 , and  12 .
- Position  2  contributes  2  A's
- Position  7  contributes  7  A's
- Position  12  contributes  12  A's

Total A's =  2  +  7  +  12  =  21

Step 5:  Calculate probability.
 Probability =  21  /  91  =  3/13

Answer: C

The most common mistake  here is miscounting the letters in MATHEMATICIAN (it has  13  letters, not  12  or  11 ) or missing one of the  3  positions where A appears.  A helpful trick: always write out each letter with its position number to avoid errors.

The letters in the word MATHEMATICIAN are written out in the following

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