Question

Prove that the total number of arrangements of the letters in the expression $x^3{y}^2{z}^4$ when written at full length is 1260.

Answer 1

There will be $3+2+4=9$ letters when the expression is written at full length.

Out of these 9 we have 3 alike of one kind, 2 alike of another and 4 alike of third kind.
Hence the required number of arrangement is
$\frac{9!\left(different\right)}{(3!)(2!)(4!)\left(alike\right)}=\frac{\mathrm{9.8.7.6.5}}{6.2}$
$\frac{\left(72\right)\left(7\right)\left(30\right)}{12}=42\times 30=1260$