Sets and Relations

Set $A$ has 5 elements.

Set $B$ has 7 elements.

In a function from $A$ to $B$, every element of $A$ must be mapped to an element of $B$. Therefore, total number of functions will be

$ =7 \times 7 \times 7 \times 7 \times 7=7^5 $

Now, number of many one functions $=$ Total functions-number of one-one functions

$ \begin{aligned} & =7^5-{ }^7 C_5 \times 5! \\ & =7^5-\frac{7!}{(7-5)!5!} \times 5! \\ & =7^5-\frac{7!}{(7-5)!}=7^5-{ }^7 P_5 \\ & \qquad \quad\left(\because{ }^n P_r=\frac{n!}{(n-r)!}\right) \\ & =7^5-{ }^7 P_5 \end{aligned} $

Total number of students $=205$

Number of students passed in Math $=70$

Number of students passed in English = 105

and number of students pass in both $=30$

Now,

Number of students passed in Math or English $=$ Number of students passed in Math + Number of student passed in English $-$ Number of students passed in both $=70+105-30=145$

So, number of students fail in both subjects $=$ Total number of students $-$ Students passed in Math or English $=205-145=60$