Question

the word geometry has eight letters. three letters are chosen at random. what is the probability that two consonants and one vowel are chosen? 0.536 0.268 0.179 0.089

Explanation:

Step1: Identify vowels and consonants in "geometry"

The word "geometry" has letters: g, e, o, m, e, t, r, y. Vowels (a, e, i, o, u) here: e, o, e (so 3 vowels). Consonants: total letters 8, so consonants = 8 - 3 = 5 (g, m, t, r, y).

Step2: Calculate combinations for choosing 2 consonants and 1 vowel

Number of ways to choose 2 consonants from 5: $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$
Number of ways to choose 1 vowel from 3: $C(3,1)=\frac{3!}{1!(3 - 1)!}=\frac{3}{1}=3$
Total favorable combinations: $10\times3 = 30$

Step3: Calculate total combinations of choosing 3 letters from 8

$C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6}{3\times2\times1}=56$

Step4: Calculate probability

Probability = $\frac{\text{Favorable combinations}}{\text{Total combinations}}=\frac{30}{56}\approx0.536$

Answer:

0.536