Question

Find number of all such function $y=f\left(x\right)$ which are onto?

• A. $243$
• B. $93$
• C. $150$
• D. None of these

Answer 1

Answer: (C)

$150$

1. For all possible functions from A to B

2. $3^5=243$ (as each value in A can get matched to any one of the elements in B)

Now we remove functions in which all match to one elements only.

3. Number of such elements $=1+1+1=3$

(One when all match to 6, one when all match to 7 and one when all match to 8)

For functions in which 2 elements are only matched.

4. $3[(\frac{5}{1})+(\frac{5}{2})+(\frac{5}{3})+(\frac{5}{4})]=90$

If we select any two numbers from $6,7$ and $8.$ Possible ways are $3$.

Now we distribute these $5$ numbers from $1$ to $5$ in different possible ways between the two numbers.

This is done by $[(\frac{5}{1})+(\frac{5}{2})+(\frac{5}{3})+(\frac{5}{4})]$.

5. Hence the total onto functions are $=243-(3+90)=150$