# Attempting a combinations question using permutations.

How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word DAUGHTER?

Let me tell my combinations approach which got me the right answer:

The given word has $3$ vowels and $5$ consonants.

Number of possible combinations = $^2C_3 \times ^3C_5$. These combinations can be arranged among themselves in $5!$ ways. Hence, number of words = $5 ! \times ^2C_3 \times ^3C_5 = 3600$

Attempt using Permutations:

There are $5$ vacant places. Thus, number of permutations possible = $2!\times 3! \times^3P_2 \times ^5P_3$. This doesn't give the right answer.

Please tell me how to approach this problem using Permutations only.

The multiplier for your permutation values is $\binom 52$ (or equally $\binom 53$), representing the choice of positions for the vowels (or consonants).
$\binom 52 \times {}^3P_2 \times {}^5P_3$
• $\binom 52 = {}^2C_5$ in your notation above. – Joffan Dec 17 '17 at 21:05