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: Count consonants and vowels

In "geometry", there are 5 consonants (g, m, t, r, y) and 3 vowels (e, o, e).

Step2: Calculate total number of ways to choose 3 letters

The total number of ways to choose 3 letters out of 8 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 8$ and $r=3$. So $C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6}{3\times2\times1}=56$.

Step3: Calculate number of ways to choose 2 consonants and 1 vowel

The number of ways to choose 2 consonants out of 5 is $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$. The number of ways to choose 1 vowel out of 3 is $C(3,1)=\frac{3!}{1!(3 - 1)!}=3$. By the multiplication - principle, the number of ways to choose 2 consonants and 1 vowel is $C(5,2)\times C(3,1)=10\times3 = 30$.

Step4: Calculate the probability

The probability $P$ is the number of favorable outcomes divided by the number of total outcomes. So $P=\frac{30}{56}\approx0.536$.

Answer:

0.536