Question Detail

In how many ways can the letters of the word "CORPORATION" be arranged so that vowels always come together.

5760
50400
2880
None of above

Answer: Option B

Explanation:

Vowels in the word "CORPORATION" are O,O,A,I,O
Lets make it as CRPRTN(OOAIO)

This has 7 lettes, where R is twice so value = 7!/2!
= 2520

Vowel O is 3 times, so vowels can be arranged = 5!/3!

= 20

Total number of words = 2520 * 20 = 50400

1. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there

Answer: Option D

Explanation:

In a group of 6 boys and 4 girls, four children are to be selected such that
at least one boy should be there.
So we can have

(four boys) or (three boys and one girl) or (two boys and two girls) or (one boy and three gils)

This combination question can be solved as

$\begin{aligned} (^{6}{C}_{4}) + (^{6}{C}_{3} * ^{4}{C}_{1}) + \\ + (^{6}{C}_{2} * ^{4}{C}_{2}) + (^{6}{C}_{1} * ^{4}{C}_{3}) \\ = \left[\dfrac{6 \times 5 }{2 \times 1}\right] + \left[\left(\dfrac{6 \times 5 \times 4 }{3 \times 2 \times 1}\right) \times 4\right] + \\\left[\left(\dfrac{6 \times 5 }{2 \times 1}\right)\left(\dfrac{4 \times 3 }{2 \times 1}\right)\right] + \left[6 \times 4 \right] \\ = 15 + 80 + 90 + 24\\ = 209 \end{aligned}$

2. In how many way the letter of the word "APPLE" can be arranged

Answer: Option C

Explanation:

Friends the main point to note in this question is letter "P" is written twice in the word.
Easy way to solve this type of permutation question is as,

So word APPLE contains 1A, 2P, 1L and 1E
Required number =
$\begin{aligned} = \frac{5!}{1!*2!*1!*1!} \\ = \frac{5*4*3*2!}{2!} \\ = 60 \end{aligned}$

3. In how many way the letter of the word "RUMOUR" can be arranged

2520
480
360
180

Answer: Option D

Explanation:

In above word, there are 2 "R" and 2 "U"
So Required number will be

$\begin{aligned} = \frac{6!}{2!*2!} \\ = \frac{6*5*4*3*2*1}{4} \\ = 180 \end{aligned}$

4. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done

Answer: Option D

Explanation:

From a group of 7 men and 6 women, five persons are to be selected with at least 3 men.
So we can have
(5 men) or (4 men and 1 woman) or (3 men and 2 woman)

$\begin{aligned} (^{5}{C}_{5}) + (^{5}{C}_{4} * ^{6}{C}_{1}) + \\ + (^{5}{C}_{3} * ^{6}{C}_{2}) \\ = \left[\dfrac{7 \times 6 }{2 \times 1}\right] + \left[\left( \dfrac{7 \times 6 \times 5}{3 \times 2 \times 1} \right) \times 6 \right] + \\ \left[\left( \dfrac{7 \times 6 \times 5}{3 \times 2 \times 1} \right) \times \left( \dfrac{6 \times 5}{2 \times 1} \right) \right] \\ = 21 + 210 + 525 = 756 \end{aligned}$

5. In how many words can be formed by using all letters of the word BHOPAL

Answer: Option D

Explanation:

Required number
$\begin{aligned} = 6! \\ = 6*5*4*3*2*1 \\ = 720 \end{aligned}$

Fernando Fidel Flores 11 years ago

what has 7 letters?
I only counted 6 consonants and 5 vowels.

mastguru 11 years ago replied

Hello Fernando,
As per solution we have written like CRPRTN(OOAIO)
Here we take it 7 letters , 6 (CRPRTN) + 1 (OOAIO), as we need to vowel always come together, this way we counted them 7. Then solved the question.
Hope it helped you..
Gabriela 11 years ago

THANK YOU SOOO MUCH, IT WAS FRO A HOMEWORK AND I WAS SO CONFUSED, I KNOW HOW TO DO COMBINATIONS AND PERMUTATIONS BUT I WASNT SURE OF WICH OF THEM TO USE AND HOW TO WRITE IT....THANKS :)