Question

A five letter word is to be formed such that the letters appearing in the odd numbered positions are taken from the letters which appear without repetition in the word MATHEMATICS. Further the letters appearing in the even-numbered positions are taken from the letters which appear with repetition in the same word MATHEMATICS. The number of ways in which the five letters word can be formed?

(a) 720
(b) 540
(c) 360
(d) None of these

Answer 1

The total number of ways to form a five-letter word from MATHEMATICS, with odd positions filled from non-repeated letters and even ones from repeated letters, is 8 x 3 x 7 x 3 x 6 = 3024 ways. Therefore, the answer is (d) None of these.

Step-by-step explanation:

To solve the problem of forming a five-letter word from the word MATHEMATICS, where letters at the odd positions must come from the non-repeated set and letters at the even positions from the repeated set, we first identify these two sets:

• Non-repeated letters: M, A, T, H, E, I, C, S
• Repeated letters: M, A, T

Since there are 8 non-repeated letters, the number of ways to choose the 1st, 3rd, and 5th positions are 8, 7, and 6 respectively (as no repetition is allowed). For the 2nd and 4th positions, which must be filled with repeated letters (M, A, T), there are 3 choices for each position.

Thus, the total number of ways to form the word is:

8 (for the 1st position) × 3 (for the 2nd position) × 7 (for the 3rd position) × 3 (for the 4th position) × 6 (for the 5th position) = 3024 ways.

The given options do not include this number, so the answer is (d) None of these.