Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps f :

Q: Wait, why do we reject the pair $(7, 7)$?

This is a classic trap! The problem states the function $f$ is one-one (injective). If we chose $f(1) = 7$ and $f(3) = 7$, then $f(1) = f(3)$. By definition, a one-one function requires that distinct inputs must have distinct outputs. Since $1 \neq 3$, their images must be different. Thus, $(7, 7)$ is invalid.

Q: Why do we use $^5P_3$ and not $^5C_3$?

Great question! We use permutations ($^5P_3$) because the function mapping is an ordered assignment. Mapping element $7$ to $2$ and $9$ to $4$ is a different function than mapping $7$ to $4$ and $9$ to $2$. Since the order of the output matters, we must use permutations.

Q: Why handle $f(1)$ and $f(3)$ first?

In combinatorics, always prioritize the most restrictive constraints. By fixing the images of $1$ and $3$ first, we reduce the complexity of the problem. Once those two are 'locked in', the remaining elements become a standard permutation problem, which is much easier to solve.

Question

Let $A = \{1, 3, 7, 9, 11\}$ and $B = \{2, 4, 5, 7, 8, 10, 12\}$. Then the total number of one-one maps $f : A 	o B$, such that $f(1) + f(3) = 14$, is :

Options:

(A) 480

(B) 240

(C) 120

(D) 180

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given two sets: $A = \{1, 3, 7, 9, 11\}$ and $B = \{2, 4, 5, 7, 8, 10, 12\}$. We aim to construct a one-one function $f: A 	o B$ subject to the specific constraint $f(1) + f(3) = 14$.

The set $A$ contains $5$ elements, and the set $B$ contains $7$ elements. A one-one function ensures that each element in $A$ maps to a unique element in $B$.

### The Constraint

To satisfy $f(1) + f(3) = 14$, we identify pairs $(x, y)$ from set $B$ such that their sum is $14$. Scanning $B = \{2, 4, 5, 7, 8, 10, 12\}$, we find the following pairs:
$(2, 12)$, $(4, 10)$, and $(7, 7)$.

### The One-One Trap

The condition that $f$ is a **one-one function** is critical. If we were to select the pair $(7, 7)$, we would have $f(1) = 7$ and $f(3) = 7$.

Since $f(1) = f(3)$, this violates the definition of an injective function. Therefore, we must exclude the pair $(7, 7)$ from our considerations.

This leaves us with two valid pairs: $(2, 12)$ and $(4, 10)$. Because the inputs $1$ and $3$ are distinct, the order within these pairs matters.

For each pair, there are $2$ possible assignments (e.g., $f(1)=2, f(3)=12$ or $f(1)=12, f(3)=2$). With $2$ valid pairs, the total number of ways to map $1$ and $3$ is:
$$2 	imes 2 = 4$$

### The Remaining Journey

After mapping $1$ and $3$, we must map the remaining elements of $A$, which are $\{7, 9, 11\}$. There are $3$ elements remaining in $A$.

Since $2$ elements from $B$ have already been used, we have $7 - 2 = 5$ elements remaining in $B$. We must map $3$ elements from $A$ to $5$ elements in $B$ injectively.

This is calculated using the permutation formula $^nP_r = \frac{n!}{(n-r)!}$:
$$^5P_3 = \frac{5!}{(5-3)!} = \frac{120}{2} = 60$$

### Final Calculation

By the **Fundamental Counting Principle**, we multiply the number of ways to satisfy the constraint by the number of ways to map the remaining elements.

The total number of such functions is:
$$4 	imes 60 = 240$$

The total number of one-one functions $f: A 	o B$ satisfying the given condition is **240**.

Animated Solution for Mathematics - Permutations and Combinations: Let $A = {1, 3, 7, 9, 11}$ and $B = {2, 4, 5, 7, 8, 10, 12}$ . Then the total number of one-one maps $f : A \to B$ , such that $f (1) + f (3) = 14$ , is :

Select Answer:

Visualized Solution

Defining the Sets

The One-One Condition

The Special Constraint

Finding Valid Pairs

The One-One Trap

Valid Mapping Cases

Mapping the Remaining Elements

Available Spots in Set B

Permutations for Remaining Elements

Calculating $^{5} P_{3}$

Total Number of Maps

Final Answer

The Sigma Insight: Fundamental Principle of Counting

Analyzing the Setup

The Constraint

The One-One Trap

The Remaining Journey

Final Calculation

Similar Questions

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Select Answer:

Visualized Solution

Defining the Sets

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The Special Constraint

Finding Valid Pairs

The One-One Trap

Valid Mapping Cases

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Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 5P3​

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Final Answer

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The One-One Trap

The Remaining Journey

Final Calculation

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