Answer to Question #117119 in Combinatorics | Number Theory for Priya

Question

In a company, ID cards have 5 digit numbers.

a) How many ID cards can be formed if repetion of the digit is allowed?

b) How many ID cards can be formed if repetion of the digit is not allowed?

Answer 1

a) If repetion is allowed, then there are 10 ways to choose each of 5 digirs. Hence, by the multiplication rule, we obtain that the total number of possible different ID cards is: $10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 10^5 = 100 000.$

b) If repetion are not allowed, then there are 10 ways to choose the first digit, 9(since we have already chosen one digit as the first) ways to choose the second digit, 8 ways to choose the third digit, 7 ways to choose the fourth digit, and 6 ways to choose the fifth digit. Therefore, by the multiplication rule, the answer is:

$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 = 30240.$