Let A={(x, y): 2 x+3 y=23, x, y ∈ ℕ} and B={x:(x, y) ∈ A}. Then the number of one-one functions from A to B is equal to _________.

Solution

Maths sir · 4 steps

List the natural-number solutions first; once you know both sets have 4 elements, it is just counting.

1. Find the pairs. From 2x + 3y = 23 we get y = (23 - 2x)/3. This is a natural number only when 23 - 2x is a positive multiple of 3, which happens for x = 1, 4, 7, 10.

   x	23 - 2x	y
   1	21	7
   4	15	5
   7	9	3
   10	3	1

2. Both sets. So A = {(1,7), (4,5), (7,3), (10,1)} has 4 elements. B collects the first coordinates, so B = {1, 4, 7, 10}, which also has 4 elements.
3. Count one-one maps. In a one-one function, no two elements of A can share an image. The first element of A has 4 choices in B, the next has 3, then 2, then 1.
4. So. The number of one-one functions is 4 × 3 × 2 × 1 = 4! = 24.

The common slip: 256. Counting every function instead of only the one-one ones gives 4⁴ = 256. Once an image is used, it cannot be used again.

The idea to keep. From a set of m elements to a set of n elements (m ≤ n), the number of one-one functions is n(n-1)…(n-m+1). When m = n, this is n!.

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