Question

Set A has elements and the set B has 6 elements, then the number of injective mappings that can be defined from A to B is :

• A. 360
• B. 15
• C. 24
• D. 1296

Answer 1

The Correct Option is A

To determine the number of injective mappings from set A to set B, we should understand what an injective function is. An injective function, or one-to-one function, maps distinct elements of set A to distinct elements of set B.

Let's suppose set A has \( n \) elements and set B has \( m \) elements. For an injective function to exist, it must be that \( n \leq m \).

Given: Set A has elements and Set B has 6 elements. To find the number of injective mappings, we use the formula for permutations because each element from A has to map to a uniquely distinct element in B.

The number of injective mappings (or permutations) of choosing \( n \) items from \( m \) items is given by the permutation formula:

\[ P(m, n) = \frac{m!}{(m-n)!} \]

In this case, assume Set A also has \( n \) elements:

\[ P(6, n) = \frac{6!}{(6-n)!} \]

Since the number of injective mappings equals 360:

\[ \frac{6!}{(6-n)!} = 360 \]

The factorial of 6 is 720, so we simplify:

\[ \frac{720}{(6-n)!} = 360 \]

Solving for \((6-n)!\):

\[ (6-n)! = \frac{720}{360} = 2 \]

By evaluating factorial values:

\( 2! = 2 \), which matches our result, implying \((6-n) = 2\).

Solve for \( n \):

\[ 6-n = 2 \]

\[ n = 4 \]

Thus, set A should have 4 elements for there to be 360 injective mappings from set A to set B.

Therefore, the number of injective mappings that can be defined from Set A to Set B is:

360