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 (3 vowels). Consonants: 8 - 3 = 5 (g, m, t, r, y).

Step2: Calculate combinations for choosing 2 consonants and 1 vowel

Combination formula: $C(n, k)=\frac{n!}{k!(n - k)!}$

• Choose 2 consonants from 5: $C(5, 2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$
• Choose 1 vowel from 3: $C(3, 1)=\frac{3!}{1!(3 - 1)!}=\frac{3}{1}=3$
• 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 = $\frac{\text{Favorable combinations}}{\text{Total combinations}}=\frac{30}{56}\approx0.536$ (Wait, no—wait, 3 vowels? Wait, "geometry" letters: g, e, o, m, e, t, r, y. Let's list again: positions 1:g, 2:e, 3:o, 4:m, 5:e, 6:t, 7:r, 8:y. So vowels: e (2), o (1), e (5) → 3 vowels. Consonants: g (1), m (4), t (6), r (7), y (8) → 5 consonants. Wait, but when choosing 3 letters, the total combinations are C(8,3)=56. Favorable: C(5,2)C(3,1)=103=30. 30/56≈0.5357≈0.536. Wait, but let me check again. Wait, maybe I miscounted vowels? Wait, vowels in English are a, e, i, o, u. So in "geometry": e, o, e → 3 vowels. Consonants: g, m, t, r, y → 5. So yes. So 30/56≈0.536.

Answer:

0.536 (corresponding to the first option)