Let A = {(x, y) : 2x + 3y = 23, x, y ∈ N} and B = {x : (x, y) ∈ A}. Then the Number of One-One Functions From A to B Is Equal To

JEE Main 2024 — Previous Year Question.

Problem

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

Key insight. Once the natural-number solutions to $2x+3y=23$ are listed out, both $A$ and $B$ turn out to have exactly the same number of elements — and a one-one function between two finite sets of equal size is automatically a bijection, so the count is simply that common size, factorial.

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Approach

Find every pair of natural numbers $(x,y)$ satisfying $2x+3y=23$ by substituting each valid value of $y$ and checking whether the resulting $x$ is a positive integer. This gives the elements of $A$ directly, and $B$ is just the set of the $x$-values appearing in those pairs. Once $|A|$ and $|B|$ are known, the count of one-one functions from $A$ to $B$ follows from the standard formula for injective functions between finite sets.

Solution

Step 1 — Find all natural-number solutions

Since $2x=23-3y$ must be even and positive, $23-3y$ must be even. As $23$ is odd, $3y$ must also be odd, which happens exactly when $y$ is odd. Checking successive odd values of $y$:

$$y=1: \quad x=\frac{23-3}2=10$$

$$y=3: \quad x=\frac{23-9}2=7$$

$$y=5: \quad x=\frac{23-15}2=4$$

$$y=7: \quad x=\frac{23-21}2=1$$

$$y=9: \quad x=\frac{23-27}2=-2 \quad (\text{invalid, not a natural number})$$

Step 2 — List the elements of A and B

$$A = \{(10,1),(7,3),(4,5),(1,7)\}$$

$$B = \{10,7,4,1\}$$

So $|A|=4$ and $|B|=4$.

Step 3 — Count the one-one functions from A to B

Since $|A|=|B|=4$, every one-one (injective) function from $A$ to $B$ is automatically also onto — it’s a bijection. The number of bijections between two sets of size 4 is:

$$4! = 4\times3\times2\times1 = 24$$

Answer

Common mistakes

• Missing a valid solution or including an invalid one. It’s easy to stop checking odd values of $y$ too early, or to forget to verify that a candidate $x$ is actually a positive integer — each of the four pairs found here should be checked by direct substitution back into $2x+3y=23$.
• Assuming |A| and |B| are different. Since $B$ is defined as the set of $x$-values from $A$’s ordered pairs, and each pair here happens to have a distinct $x$-value, $|B|$ ends up equal to $|A|$ — this isn’t automatic in general (if two pairs shared the same $x$, $B$ would be smaller), so it’s worth explicitly listing $B$’s elements rather than assuming its size.
• Using the general injective-function formula unnecessarily. For $|A|=m\le|B|=n$ in general, the count of one-one functions is $\dfrac{n!}{(n-m)!}$ — but when $m=n$ exactly, this reduces to simply $n!$, since $(n-m)!=0!=1$.

Practise next

• Let $A=\{(x,y):2x+3y=17,\ x,y\in\mathbb N\}$ and $B=\{x:(x,y)\in A\}$. Find the number of one-one functions from $A$ to $B$.