Question

How many 3-digit numbers can we make using the digits 1, 2, 3, 4, 5, 6 without repetitions? How about with repetitions (meaning different digits can hold the same number; e.g., 223 and 444 are valid 3-digit numbers in the with repetitions case)?

Answer 1

we have the following digits 1,2,3,4,5,6

(A) Without repetitions

We have to make a 3 digit numbers

So, for first digit, we can select any number of out given six numbers = 6 ways to fill first digit

now, we have only 5 digits after filling the first place, so we can select any number of out remaining five number = 5 ways to select

And, for third digit, we have only numbers to select = 4 ways to select the third digit

So, total number of ways of selecting 3 digit number without repetitions = 6*5*4 = 120 different ways

(B) with repetitions

We have to make a 3 digit numbers and with repetition, we can select any number out of six numbers at any place.

So, we can select first digit in 6 different ways

similarly, we can select the second and third digit in 6 different ways each because repetition is allowed.

So, total number of ways of selecting 3 digit number with repetitions= 6*6*6 = 216 different ways