6
Q:

# How many arrangements can be made out of the letters of the word COMMITTEE, taken all at a time, such that the four vowels do not come together?

 A) 216 B) 45360 C) 1260 D) 43200

Explanation:

There are total 9 letters in the word COMMITTEE in which there are 2M's, 2T's, 2E's.

$\inline \therefore$ The number of ways in which 9 letters can be arranged = $\inline \frac{9!}{2!\times 2!\times 2!}$ = 45360

There are 4 vowels O,I,E,E in the given word. If the four vowels always come together, taking them as one letter we have to arrange 5 + 1 = 6 letters which include 2Ms and 2Ts and this be done in $\inline \frac{6!}{2!\times 2!}$ = 180 ways.

In which of 180 ways, the 4 vowels O,I,E,E remaining together can be arranged in $\inline \frac{4!}{2!}$ = 12 ways.

$\inline \therefore$ The number of ways in which the four vowels always come together = 180 x 12 = 2160.

Hence, the required number of ways in which the four vowels do not come together = 45360 - 2160 = 43200

Q:

Some children goto ice-cream shop. 9 flavours are available there. Each child takes a cone with two different flavours. No two children take same combination and they have taken all such possible combinations. How many children went to ice cream shop ?

 A) 28 B) 56 C) 44 D) 36

Explanation:

Given there are 9 flavours of ice creams.
Each child takes the combination of two flavours and no two children will have the same combination
This can be done by 9C2 ways i.e children.
Number of children = 9C2 = 9 x 8 / 2 = 36.

2 22
Q:

In how many ways can the letters of the word 'LEADER' be arranged ?

 A) 360 B) 420 C) 576 D) 220

Explanation:

No. of letters in the word = 6
No. of 'E' repeated = 2
Total No. of arrangement = 6!/2! = 360

1 22
Q:

In how many different ways could couples be picked from 6 men and 9 women ?

 A) 26 B) 54 C) 52 D) 28

Explanation:

Number of mens = 8
Number of womens = 5

Different ways could couples be picked = 6c1 x 9c1 = 9 x 6 = 54 ways.

2 43
Q:

How many 6-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 and 7 so that the digits should not repeat and the second last digit is even ?

 A) 521 B) 720 C) 420 D) 225

Explanation:

Let last digit is 2
when second last digit is 4 remaining 4 digits can be filled in 120 ways, similarly second last digit is 6 remained 4 digits can be filled in 120 ways.
so for last digit = 2, total numbers=240

Similarly for 4 and 6
When last digit = 4, total no. of ways =240
and last digit = 6, total no. of ways =240
so total of 720 even numbers are possible.

3 29
Q:

How many 4-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed ?

 A) 5040 B) 2525 C) 2052 D) 4521