Question

Find the number of ways in which : (a) a selection (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'.

Answer 1

Given word PROPORTION

$P-2,R-2,0-3,T,I,N,$

$\left(i\right)$ Words with all distinct letters $=6_{c4}=15$ways

Arrangement=$6_{p4}=360$ ways

$\left(ii\right)$ One letter repeat twice

Selection $=3_{c1}\times 5_{c2}${i.e one from $P/R/O$ & $2$ from the $5$}

Arrangement $=3_{c1}\times 5_{c2}\times \frac{4!}{2!}=360$

$\left(iii\right)$Two letter repeated twice

Selection $=3_{c2}=3$ [i.e two from $P/R/O$]

Arrangement $=3_{c2}\times \frac{4!}{2!.2!}=3\times 6=18$

$\left(iv\right)$ Three letter some

Selection $=1_{c1}\times 5_{c1}=5$ [i.e one from orther $3$ all ${}^{\prime }0"$ ]

Arrangement$=5\times \frac{4!}{3!}=20$

$\therefore$ Total selection $=15+30+3+5$

$=53$ ways

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

$=758$ ways