The number of ways of arranging all the letters of the word "COMBINATIONS" around a circle so that no two vowels come together is
Step 1: Identify Consonants and Vowels
The word "COMBINATIONS" consists of 11 letters: - Vowels: (5 vowels) - Consonants: (7 consonants) To ensure that no two vowels are adjacent, we first arrange the consonants in a circular arrangement.
Step 2: Arrange Consonants in a Circle
Since circular permutations of distinct objects is given by , the consonants can be arranged in:
Step 3: Placing Vowels in Gaps
Once the consonants are placed in a circle, they create 7 gaps. The 5 vowels must be placed in these gaps. The number of ways to choose 5 gaps from 7 is: Since vowels include repetitions, we divide by factorials of repeated letters:
Final Answer:
With respect to the roots of the equation , match the items of List-I with those of List-II.
A man has 7 relatives, 4 of them are ladies and 3 gents; his wife has 7 other relatives, 3 of them are ladies and 4 gents. The number of ways they can invite them to a party of 3 ladies and 3 gents so that there are 3 of man's relatives and 3 of wife's relatives, is
Let be a matrix with positive integers as its elements. The elements of are such that the sum of all the elements of each row is equal to 6, and .