Question

Number of one-one functions from A to B where n(A)= 4 and n(B)= 5.
a). 4
b). 5
c). 120
d). 90

Answer 1

Hint: Type of questions is based on the function topic more precisely on the classification of Function. As there are five types of functions
i). one-one function
ii). many one function
iii). onto function
iv). into function
v). Bijective function

Complete step-by-step solution:
One –one function is also known as injective function. A one-one function is a function in which for one input we get one output. As to find one –one function we had a direct formula through which we can easily find the one-one function of ‘A’ to ‘B’.
The formula is $${}^n{P}_m$$ only if $n\ge m$
In which ‘n’ and ‘m’ are the number of elements of B and A respectively. and where p is for permutation, which is further solved as ${\displaystyle \frac{n!}{(n-m)!}}$
Moving further with our question where
n(A)= number of elements of A= 4= m
n(B)= number of elements of B= 5= n
So by comparing it with above formula ‘m’ = 4 and ‘n’= 5
On solving we will get
$$\begin{array}{rl}& {=}^n{P}_m\\ & {=}^5{P}_4\\ & ={\displaystyle \frac{5!}{(5-4)!}}\\ & ={\displaystyle \frac{5!}{1!}}\\ & ={\displaystyle \frac{5\times 4\times 3\times 2\times 1}{1}}\\ & =120\end{array}$$
Hence, the number of one-one functions are 120. So the answer is 120, i.e. option C.

Note: while solving keep in mind that ‘n’ is the number of elements in ‘B’ and ‘m’ is the number of elements in ‘A’. Moreover this formula is only valid when n is greater than or equal to m $n\ge m$ . If it comes out to be less than m than the answer would be zero (0).