calculating probabilities of letter choices\nthe word geometry has eight letters. three letters are chosen…

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 number of ways to choose 2 consonants out of 5

Using combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 5$ and $k=2$. So $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$.

Step3: Calculate number of ways to choose 1 vowel out of 3

Using combination formula with $n = 3$ and $k = 1$. So $C(3,1)=\frac{3!}{1!(3 - 1)!}=\frac{3}{1}=3$.

Step4: Calculate number of ways to choose 3 letters out of 8

Using combination formula with $n = 8$ and $k = 3$. So $C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6}{3\times2\times1}=56$.

Step5: Calculate probability

The number of favorable cases is the product of number of ways to choose 2 - consonants and 1 - vowel, which is $C(5,2)\times C(3,1)=10\times3 = 30$. The probability $P=\frac{30}{56}\approx0.536$.

Answer:

0.536