Question

Find the number of $7$ lettered words each consisting of $3$ vowels and $4$ constant which can be formed using the letters of the word " DIFFERENTIATION".

Answer 1

If we assume that each vowel & consonant can only occur only one time, Some vowels and consonants appear multiple times.

$1.$ Three same vowel (III only) : $1$

$2.$ Two same vowels (either II or EE) and one unique vowel. : $2C_1\times 3C_1\times 3C^2=18$

$3.$ Three unique vowels : $3!\times 4C_3=24$

$\therefore$ There are total of $1+18+24=43$ possible combinations.

$1.$ Two times same consonants : $3C_2\times 4C_2=18$

$2.$ Two same consonants and two unique consonants. $2!\times 3C_1\times 4C_2\times 4C_2=216$

$3.$ Four unique consonants. $4!\times 5C_4=120$

$\therefore \text{Total possibilities}=18+216+120=354.$

Selecting $3$ position to place the vowels,, the total no. of words $43\times 354\times 7C=532770$