Question

Question 9 How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if

(i) 4 letters are used at a time, (ii) all letters are used at a time,

(iii) all letters are used but first letter is a vowel?

Class X1 - Maths -Permutations and Combinations Page 148

Answer 1

in the word MONDAY. all are different letters
(i) out of 6 different letters 4 letters can be selected in ⁶P₄ ways.
then, required number of word = 6!/(6-4)!
= 6 × 5 × 4 × 3 × 2!/2!
= 6 × 5 × 4 × 3
= 360

(ii) all letters are used at a time.
number of ways taking 6 letters at a time = ⁶P₆ ways
then, required number of ways = ⁶P₆
= 6!/(6-6)!
= 6 × 5 × 4 × 3 × 2 × 1
= 720 ways .


(iii) in word ' MONDAY ' there are two vowels are O and A .
first letter can be selected by 2 ways .
number of ways taking 5 different letters from remaining 5 letters = ⁵P₅ ways.
= 5!/(5 - 5)!
= 5 × 4 × 3 × 2 × 1
= 120 ways.
then, required number of ways = 2 × 120 ways
= 240 ways .

Answer 2

(i) 360 ways

(ii) 720 ways

(iii) 240 ways

Step-by-step explanation:

No. of letters in the word MONDAY = 6

(i) When 4 letters are used at a time.

Then, the required number of words:

= 6P4

= $\frac{6!}{2!} = 6 * 5* 4* 3 = 360$

Therefore, there are 360 ways.

(ii) When all letters are used at a time. Then the required number of words

= 6P6 = 6!

= 720

Therefore, there are 720 ways.

(iii) All letters are used but first letter is a vowel.

So the first letter can be either A or O.

So there are 2 ways to fill the first letter & remaining places can be filled in 5P5 ways.

∴ The required number of words:

= 2 x 5P5

= 2 x 5! =240

Therefore, there are 240 ways.