5. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together? | |
A. 42000 | B. 48000 |
C. 50400 | D. 47200 |
Answer: Option C
Explanation:
The word 'CORPORATION' has 11 letters. It has the vowels 'O','O','A','I','O' in it and these 5 vowels should always come together. Hence these 5 vowels can be grouped and considered as a single letter. that is, CRPRTN[OOAIO].
Hence we can assume total letters as 7. But in these 7 letters, 'R' occurs 2 times and rest of the letters are different.
Number of ways to arrange these
letters
$=\dfrac{7!}{2!}=\dfrac{7×6×5×4×3×2×1}{2×1}=2520$
In the 5 vowels [OOAIO], 'O' occurs 3 and rest of the vowels are different.
Number of ways to arrange these vowels among themselves $=\dfrac{5!}{3!}=\dfrac{5×4×3×2×1}{3×2×1}=20$
Hence, required number of ways
$=2520×20=50400$
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Solution
The correct option is A 50400
In the word 'CORPORATION', we’ll
treat the vowels OOAIO as a single letter. Thus, we have CRPRTN [OOAIO].
This has 7 [6 + 1] letters of which R occurs 2 times and the rest are different. Number of ways of arranging these letters
=7!2!!=2520
Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged
In
5!3!=20 ways
Therefore, Required number of ways
=[2520×20]=50400
In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
A. 810
B. 1440
C. 2880
D. 50400
E. 5760
Answer: Option D
Solution[By Examveda Team]
In the word 'CORPORATION', we treat the vowels OOAIO as one letter.
Thus, we have CRPRTN [OOAIO].
This has 7 [6 + 1] letters of which R occurs 2 times and the rest are different.
Number of ways arranging these letters = $$\frac{{7!}}{{2!}}$$ = 2520
Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in $$\frac{{5!}}{{3!}}$$ = 20 ways
∴ Required number of ways = [2520 x 20] = 50400
Question Detail
- 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
Similar Questions :
1. A box contains 4 red, 3 white and 2 blue balls. Three balls are drawn at random. Find out the number of ways of selecting the balls of different colours
- 12
- 24
- 48
- 168
Answer: Option B
Explanation:
This question seems to be a bit typical, isn't, but it is simplest.
1 red ball can be selected in 4C1 ways
1 white ball can be selected in 3C1 ways
1 blue ball can be selected in 2C1 ways
Total number of ways
= 4C1 x 3C1 x 2C1
= 4 x 3 x 2
= 24
Please note that we have multiplied the combination results, we use to add when
their is OR condition, and we use to multiply when there is AND condition, In this question it is AND as
1 red AND 1 White AND 1 Blue, so we multiplied.
2. Evaluate permutation equation
\begin{aligned} ^{59}{P}_3 \end{aligned}
- 195052
- 195053
- 195054
- 185054
Answer: Option C
Explanation:
\begin{aligned}
^n{P}_r = \frac{n!}{[n-r]!} \\
^{59}{P}_3 = \frac{59!}{[56]!} \\
= \frac{59 * 58 * 57 * 56!}{[56]!} \\
= 195054
\end{aligned}
3. In how many words can be formed by using all letters of the word BHOPAL
- 420
- 520
- 620
- 720
Answer: Option D
Explanation:
Required number
\begin{aligned}
= 6! \\
= 6*5*4*3*2*1 \\
= 720
\end{aligned}
4. Evaluate permutation equation
\begin{aligned} ^{75}{P}_2\end{aligned}
- 5200
- 5300
- 5450
- 5550
Answer: Option D
Explanation:
\begin{aligned}
^n{P}_r = \frac{n!}{[n-r]!} \\
^{75}{P}_2 = \frac{75!}{[75-2]!} \\
= \frac{75*74*73!}{[73]!} \\
= 5550
\end{aligned}
5. How many words can be formed by using all letters of TIHAR
- 100
- 120
- 140
- 160
Answer: Option B
Explanation:
First thing to understand in this question is that it is a permutation question.
Total number of words = 5
Required number =
\begin{aligned}
^5{P}_5 = 5! \\
= 5*4*3*2*1 = 120
\end{aligned}
Read more from - Permutation and Combination Questions Answers
-
Fernando Fidel Flores 8 years ago
what has 7 letters?
I only counted 6 consonants and 5 vowels.mastguru 8 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 9 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 :]
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