Question

How many ways can someone pick (from the English language) a vowel, a consonant, and a single digit?

Answer 1

1050 ways

Explanation:

If I'm reading the question correctly, we're looking to pick from the population of all 26 letters in the English alphabet and 10 digits three things - 1 vowel, 1 consonant, and 1 digit.

There are 5 vowels: a, e, i, o, u. And so there are 5 ways to pick a vowel.

There are therefore `26-5=21` consonants. And so there are 21 ways to pick a consonant.

And there are 10 digits, and so there are 10 ways to pick a digit.

That gives:

`5xx21xx10=1050` ways.

Let's take this one step further. How many different ways can we pick 3 things from the population of 36 things? This is a combination question (we don't care about the order in which we pick the three things):

`C_(n,k)=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

`C_(36,3)=(36!)/((3!)(36-3)!)=(36!)/(3!33!)=(36xx35xx34xx33!)/(6xx33!)=7140`

And so the probability of picking 1 vowel, 1 consonant, and 1 digit when picking 3 things from our population of letters and digits is:

`1050/7140=5/34~=17.4%`