Question

21. ID Cards How many different ID cards can be made if there are 6 digits on a card and no digit can be used more than once?

Answer 1

We have to find the number of ID cards that can be made if there are 6 digits on a card and no digit can be used more than once.

We know that if we have a group of n objects and we need to know the number of permutations of r objects taken from those n objects, we use the permutation formula, i.e., n$\mathbb{P}$r = n!/(n-r)!, where, x! = x(x-1)(x-2)(x-3).......1. --------------(1)

Thus, here we can choose from 10 digits which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Thus, we have to look for number of orders for choosing 6 digits from 10 digits, i.e., the number of permutations of 6 digits taken from these 10 digits.

Thus, here, n = 10 and r = 6. Thus, putting these values in (1), we get,

Number of permutations = 10$\mathbb{P}$6 = 10!/(10-6)! = 10!/4! = (10*9*8*7*6*5*4!)/4! = 10*9*8*7*6*5 = 151200.

Thus, number of permutations = Number of orders = 151200 .

Thus, there are 151200 different ID cards that can be made if there are 6 digits on a card and no digit can be used more than once.