If a set P contains 5 elements and the set Q contains 8 elements, then the number of one-one functions from P to Q is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \({}^{8}C_{5} \times 5!\)
In a one-one function from P to Q, the 5 elements of P must go to 5 different elements of Q.
The first element of P has 8 choices, the second 7, the third 6, the fourth 5 and the fifth 4:
\(8 \times 7 \times 6 \times 5 \times 4 = 6720\)
This is the same as choosing the 5 images out of 8 and then arranging them among the 5 elements:
\( \displaystyle {}^{8}C_{5} \times 5! = \frac{8!}{5!\,3!} \times 5! = \frac{8!}{3!} = 6720 \)
Why the other options are incorrect
Option (1) \({}^{8}C_{5}\) — Incorrect. It only chooses which 5 elements of Q are used (56 ways) and leaves out the 5! ways of assigning them.
Option (3) \(5^8\) — Incorrect. This would count all functions from an 8-element set to a 5-element set, the wrong direction.
Option (4) \(8^5\) — Incorrect. This counts all functions from P to Q, including those that send two elements to the same image.
Note: NTA dropped this question. For the benefit of students and to give them additional practice, the question has been suitably modified so that it has one clear answer.