Question

Functions are formed from the set $A = \{a_1, a_2, a_3\}$ to another set $B = \{b_1, b_2, b_3, b_4, b_5\}$. If a function is selected at random, the probability that it is a one-one function is

• A. 1/2
• B. 13/25
• C. 3/5
• D. 12/25

Hint:

For a function from a set A with $n$ elements to a set B with $m$ elements: - The total number of functions is $m^n$. - The number of one-one (injective) functions is $^mP_n = \frac{m!}{(m-n)!}$ (if $n \le m$, otherwise 0).

Answer 1

The Correct Option is D

Step 1: Calculate the total number of possible functions from A to B.
Let $|A|=n=3$ and $|B|=m=5$.
A function from A to B maps each of the 3 elements in A to one of the 5 elements in B.
For the first element $a_1$, there are 5 choices in B.
For the second element $a_2$, there are 5 choices in B.
For the third element $a_3$, there are 5 choices in B.
The total number of functions is $5 \times 5 \times 5 = 5^3 = 125$.

Step 2: Calculate the number of one-one (injective) functions from A to B.
A function is one-one if each element in A maps to a unique element in B. No two elements in A can map to the same element in B.
For the first element $a_1$, there are 5 choices in B.
For the second element $a_2$, since its image must be different from the image of $a_1$, there are only 4 remaining choices in B.
For the third element $a_3$, its image must be different from the first two, so there are 3 remaining choices in B.
The number of one-one functions is $5 \times 4 \times 3 = 60$.
This can also be calculated using the permutation formula $^mP_n = ^5P_3 = \frac{5!}{(5-3)!} = 60$.

Step 3: Calculate the probability of selecting a one-one function.
Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.
Probability(one-one) = $\frac{\text{Number of one-one functions}}{\text{Total number of functions}}$.
Probability = $\frac{60}{125}$.

Step 4: Simplify the probability.
We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 5.
$\frac{60 \div 5}{125 \div 5} = \frac{12}{25}$.