Alphabetical combinatorics

Probability Level 3

Out of $7$ consonants and $4$ vowels ,how many words scan be made each containing $3$ consonants and $2$ vowels ?

The answer is 25200.

2 solutions

Maggie Miller
Aug 8, 2015

Assume that a word cannot repeat letters.

There are ${7 \choose 3}{4 \choose 2}=210$ ways to choose the letters for the word. Given the letters, there are $5!=120$ ways to arrange them. Therefore, there are $210\cdot 120=\boxed{25200}$ such words.

I find this problem a bit misleading as I am sure that all 25200 combinations would not actually make up words, such as ovit, at least in english. The correct phrasing would be "how many combinations can be made each containing 3 consonants and 2 vowels ?"

Scott Ripperda - 5 years, 10 months ago

In mathematics/Computer science, "word" means any finite (sometimes infinite) string of of letters out of a given list (alphabet). There's some ambiguity because of the English definition, but since that one wouldn't make sense here it's clear which one they meant.

Maggie Miller - 5 years, 10 months ago

A Former Brilliant Member
Mar 28, 2018

We can choose $3$ consonants from the $7$ consonants: $_7C_3=35$

We can choose $2$ vowels from the $4$ vowels: $_4C_2=6$

Then arrange the $5$ letters: $5!=120$

So the number of words is $35(6)(120)=\boxed{25200}$

Alphabetical combinatorics

The answer is 25200.

2 solutions

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