Combination with Replacement Calculator

Combination with replacement calculator calculates the total number of ways to select objects where the order does not matter, and the same object can be selected multiple times.

Combination with Replacement Formula

combination with replacement formula is

C^R(n, r) = (n + r - 1)! / r!( n - 1)!

where
n = number of objects
r = sample size (how many times of objects to be chosen)

Combination with Replacement Example

Unlike a combination that does not allow replacement,
let’s consider an example where you have 4 different colors of marbles in a bag:
white, black, blue, and red.
You want to pick 2 marbles out of the 4,
after picking the first marble (e.g., black),
you have only 3 choices left in the bag to select from(white, blue, red),
which means you do not replace the marbles back into the bag.

The combination is calculated as follows:

4C2 = 4! / 2!2! = 6

The specific combinations without replacement are:
   1.White, Black
   2.White, Blue
   3.White, Red
   4.Black, Blue
   5.Black, Red
   6.Blue, Red

Now, consider the scenario with replacement.
After picking the first marble (e.g., black),
you put it back into the bag before picking the second one.
This means you still have 4 choices for the second pick.

In this case, the number of combinations is calculated using combination with replacement formula:

C^R(4,2)= (4+2-1)! / 2!(4-1)! = 10

The specific combinations with replacement are:
   1.White, White
   2.White, Black
   3.White, Blue
   4.White, Red
   5.Black, Black
   6.Black, Blue
   7.Black, Red
   8.Blue, Blue
   9.Blue, Red
   10.Red, Red