Question

The number of permutations by taking all letters and keeping the vowels of the word ‘COMBINE’ in odd places is equal to:
A. 96
B. 144
C. 512
D. 576

Answer 1

Hint: In this question we need to find the method of rearrangement of the word COMBINE, keeping vowels only at odd places. Since there are 3 vowels, and 4 places to keep them, we have essentially \[{}^4{P_3}\] ways to arrange the vowels, and for remaining letters we have 4 places and 4 letters so we can simply put them anywhere so their permutation will be given by \[{}^4{P_4}\] and the total is given by the product of the two permutations.

Formula Used:
The general formula for permutation is given by:
\[\dfrac{n!}{\left ( n-r \right )!}\]

Complete Step-By-Step Solution: The given word is COMBINE.
The vowels in the word COMBINE are O, I, E that is 3 vowels.
Other words that are in the odd places are C, M, B, N which are at 1, 3, 5, 7. That means there are 4 other words.
Therefore, the required number of ways in which the word COMBINE can be arranged is written using the above formula:
\[{}^{4}P_{3}\times 4!\]
Solving above equation we can write that
\[n = 24 \times 24\]
On multiplying the above equation we get,
n=576
Therefore, the number of permutations by taking all the letters and keeping the vowels of the word COMBINE in odd places will be equal to 576.

Hence, Option D is the correct answer.

Note: Permutation is a method in which the sets or patterns are decided on the basis of the order of the data that is already given. It is used to arrange alphabets, number, positions, digits etc. It is important to remember that permutation is different from combination. In permutation, the objects are arranged in a definite pattern. But in case of combination, the order in which the objects are arranged is not taken into consideration. Also, in combination, the objects are selected not arranged. In permutation, the objects can be repeated as they are different.