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:
One Condition
Therefore, No. of ways = 6*1 = 6 ……(1)
Case 2:
Two conditions
Therefore, No. of ways = 6*2 = 12 ……(2)
Case 3:
Three Conditions
Therefore, No. of ways = 6*3 = 18 ……(3)
Case 4:
Four Conditions
Therefore, No. of ways = 6*4 = 24 ……(4)
Case 5:
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