Question

How many 4 letter words can be formed using the letters of the word PROPORTION.

Answer 1

Given word is PROPORTION

we have $2P^{\prime }s,2R^{\prime }s,3o^{\prime }s$ and 1 T, 1 and N

1. word with four distinct letters

Total letters (distinct) $=6$

$\therefore {}^6{P}_4\times 4!=360$ ways

2. Words with exactly a letter repeated twice ${}^3{P}_1=3$ way

The other two letters can be selected in $5P_2=10$ ways

Now each combination can be arranged in $4!\times 2!=12$ ways

Total no. = $3\times 10\times 12=360$

3. Words with exactly two distinct letters repeated twice

Two letters out of $3$ repeating = ${}^3{P}_2$ ways

Now each combination can be arranged in $4!\times 2!\times 2!=6$ ways

Total = $3\times 6=18$

4. Words with exactly a letter repeated

We have only $O,$

${\therefore }^5{P}_1=5$

each combination can be arranged in $4!/3!=4$

Total = $1\times 5\times 4=20$

$\therefore$ Total arrangement $=360+360+18+20$

$=758$ ways