The number of ways of arranging all the letters of the word "COMBINATIONS" around a circle so that no two vowels come together is
\( \frac{7! \times ^8P_5}{(2!)^3} \)
Step 1: Identify Consonants and Vowels
The word "COMBINATIONS" consists of 11 letters: - Vowels: \( O, I, A, I, O \) (5 vowels) - Consonants: \( C, M, B, N, T, N, S \) (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 \( n \) distinct objects is given by \( (n-1)! \), the consonants can be arranged in: \[ (7-1)! = 6! \]
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: \[ \text{Ways to place vowels} = ^7P_5 \] Since vowels \( O, I, A, I, O \) include repetitions, we divide by factorials of repeated letters: \[ \frac{7!}{(2!)^4} \]
Final Answer: \[ \frac{7!}{(2!)^4} \]
With respect to the roots of the equation \( 3x^3 + bx^2 + bx + 3 = 0 \), 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
The following graph is obtained for the adsorption of a gas on the surface of a catalyst. The values of k and n are respectively
Observe the following reactions (not balanced):
Cl2 + NaOH → NaCl + X + H2O
Cl2 + NaOH → NaCl + Y + H2O
Which of the following is not correct?
(1) XeO2 is a colorless explosive gas
(2) SO2 is highly soluble in water
(3) Noble gases have very low boiling points
(4) The boiling point of sulphur is more than that of oxygen