Let A={1,2,3,4,5} and B={1,2,3,4,5,6}. Then the number of functions f: A → B satisfying f(1)+f(2)=f(4)-1 is equal to __________.

Solution

Maths sir · 4 steps

Once f(1) and f(2) are chosen, f(4) is fixed. Count those pairs, then let f(3) and f(5) roam free.

1. Rewrite the condition. The condition says f(4) = f(1) + f(2) + 1. So f(4) is decided completely by f(1) and f(2), and it must still lie in B, that is, f(4) ≤ 6.

   f(1) + f(2) + 1 ≤ 6 ⇒ f(1) + f(2) ≤ 5

2. Count the pairs. Both values are at least 1. Sum 2 gives 1 pair, sum 3 gives 2, sum 4 gives 3, sum 5 gives 4. That is 1 + 2 + 3 + 4 = 10 pairs, and each fixes exactly one f(4).
3. Free elements. The condition does not involve f(3) or f(5), so each can be any of the 6 elements of B. That gives 6 × 6 = 36 choices.
4. So. The number of functions is 10 × 36 = 360.

The common slip: 10. Forgetting that f(3) and f(5) each have 6 free choices gives only 10, not 360.

The idea to keep. In function-counting questions, first let the condition fix whatever it can, count only the genuine choices, then multiply by |B| for every element the condition leaves alone.

Answer from the NTA final answer key. Solution by PrepWiser.