Question

The letters of the word 'LOGARITHM' are arranged at random. The probability that arrangement starts with vowel and end with consonant is ____________.

Options

• ${\displaystyle \frac{1}{9}}$

• ${\displaystyle \frac{7!}{9!}}$

• ${\displaystyle \frac{18}{9!}}$

• ${\displaystyle \frac{1}{4}}$

Answer 1

The letters of the word 'LOGARITHM' are arranged at random. The probability that arrangement starts with vowel and end with consonant is ${\displaystyle \underline{\frac{1}{4}}}$.

Explanation:

Given, word 'LOGARITHM'

Total number of arrangements = 9!

Here, vowel (O, A, I)

Consonant (L, G, R, T, H, M)

Total number of ways when starts with vowel and end with consonant is ${\displaystyle {}^3{\text{C}}_1\times {}^6{\text{C}}_1\times 7!}$

∴ Required probability = ${\displaystyle \frac{{}^3{\text{C}}_1\times {}^6{\text{C}}_1\times 7!}{9!}=\frac{1}{4}}$