Let R=a,b,c,d,e and S=1,2,3,4.

Question

Let $R={a,b,c,d,e}$ and $S={1,2,3,4}$. Total number of onto functions $f:R	o S$
such that $f(a)
e 1$, is equal to \_\_\_\_\_\_\_\_\_\_\_\_\_\_.

Answer: 180

Accepted Answer

Total number of onto functions

$\begin{gathered} =\frac{5!}{3!2!}	imes 4! \ =\frac{5	imes 4}{2}	imes 24=240 \end{gathered}$

When $f(a)=1$, number of onto functions

$\begin{gathered} =4!+\frac{4!}{2!2!}	imes 3! \ =24+36=60 \end{gathered}$

So, required number of onto functions

$=240-60=180$

Solutions · 1