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 repetitions in the same word MATHEMATICS. In how many different ways the five letter word can be formed –

• A. 390
• B. 600
• C. 540
• D. 450

Answer 1

Answer: (C)

540

Answer 2

In word MATHEMATICS

H, E, C, I and S – without repetition

M, A, T – occurs twice 5 letters can be placed on 3 places in ⁵P₃ ways.

Again even places 2^nd & 4^th position can be filled by the three letter M, A & T

Even places can be filled in two ways

(1) Choose 1 letter from 3 given letters M, A & T

³C₁ ways

(2) Choose 2 letter from 3 given letters M, A & T and arrange them in 2! ways

³C₁ × 2! ways

Total ways ³C₁ + ³C₁ × 2! = 9 ways

Required number of ways = ⁵P₃ × 9 = 540 ways.