Why did we use P&C here ?!.. Couldn't we use multiplication theorem, like for finding P(X=0), couldn't we write 4/52* 4/52 ?!..

Why did we use P&C here ?!.. Couldn't we use multiplication theorem, like for finding P(X=0), couldn't we write 4/52* 4/52 ?!.. Lang. Solution X Oenolc cards X 0.1

We can use permutations and combinations to help us answer more complex probability questions, see the example below. You would realise how cumbersome and unfeasible it becomes to use simple Multiplication for problems involving tremendous calculations.

Example 1

A 4 digit PIN is selected. What is the probability that there are no repeated digits?

There are 10 possible values for each digit of the PIN (namely: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), so there are 10 × 10 × 10 × 10 = 10
⁴ = 10000 total possible PINs.

To have no repeated digits, all four digits would have to be different, which is selecting without replacement. We could either compute 10 × 9 × 8 × 7, or notice that this is the same as the permutation
₁₀P₄ = 5040.

The probability of no repeated digits is the number of 4 digit PINs with no repeated digits divided by the total number of 4 digit PINs. This probability is

$\frac{{}_{10}{P}_4}{{10}^4}=\frac{5040}{10000}=0.504$