The total number of functions, f: {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to

(A) 60
(B) 90
(C) 108
(D) 126

Solution:

Tip for solving this question:

First, find f(3) at different values.

Calculate the number of ways at different values of f(3) to find the total number of functions

Step 1 of 2:

Given, f: {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6}

Total number of ways = {No. of ways of selecting f(1), f(2), f(3)}

*{No. of ways of selecting f(4)}

Now, No. of ways of selecting f(4) = 6

Here f(3) can be 2, 3, 4, 5, 6

Also, Given f(1) + f(2) = f(3)

Different cases at different values of f(3) are

Case 1:

$\begin{gathered} f(3) = 2 \hfill \\ \Rightarrow f(1) + f(2) = 2 \hfill \\ \end{gathered}$

$\Rightarrow f(1) = f(2) = 1$

One Condition

Therefore, No. of ways = 6*1 = 6 ……(1)

Case 2:

$\begin{gathered} f(3) = 3 \hfill \\ \Rightarrow f(1) + f(2) = 3 \hfill \\ \end{gathered}$

$\Rightarrow \left. \begin{gathered} f(1) = 1\& f(2) = 2 \hfill \\ f(1) = 2\& f(2) = 1 \hfill \\ \end{gathered} \right\}$

Two conditions

Therefore, No. of ways = 6*2 = 12 ……(2)

Case 3:

$\begin{gathered} f(3) = 4 \hfill \\ \Rightarrow f(1) + f(2) = 4 \hfill \\ \end{gathered}$

$\Rightarrow \left. \begin{gathered} f(1) = f(2) = 2 \hfill \\ f(1) = 1\& f(2) = 3 \hfill \\ f(1) = 3\& f(2) = 1 \hfill \\ \end{gathered} \right\}$

Three Conditions

Therefore, No. of ways = 6*3 = 18 ……(3)

Case 4:

$\begin{gathered} f(3) = 5 \hfill \\ \Rightarrow f(1) + f(2) = 5 \hfill \\ \end{gathered}$

$\Rightarrow \left. \begin{gathered} f(1) = 2\& f(2) = 3 \hfill \\ f(1) = 3\& f(2) = 2 \hfill \\ f(1) = 1\& f(2) = 4 \hfill \\ f(1) = 4\& f(2) = 1 \hfill \\ \end{gathered} \right\}$

Four Conditions

Therefore, No. of ways = 6*4 = 24 ……(4)

Case 5:

$\begin{gathered} f(3) = 6 \hfill \\ \Rightarrow f(1) + f(2) = 6 \hfill \\ \end{gathered}$

$\Rightarrow \left. \begin{gathered} f(1) = f(2) = 3 \hfill \\ f(1) = 2\& f(2) = 4 \hfill \\ f(1) = 4\& f(2) = 2 \hfill \\ f(1) = 1\& f(2) = 5 \hfill \\ f(1) = 5\& f(2) = 1 \hfill \\ \end{gathered} \right\}$

Five Conditions

Therefore, No. of ways = 6*5 = 30 ……(5)

Step 2 of 2:

From (1), (2), (3), (4), (5)

Total no. of ways = 6 + 12 + 18 + 24 + 30 = 90

i.e. Total no. of functions = 90

Final Answer:

Hence, Option (B) is correct.

JEE Main 2022 July 25th Shift 1 Mathematics Question Paper and Solutions