A little bit different version to understand same.

The formula for combinations without repetition is given by the standard combination formula:

nCr = n! / (r! * (n - r)!)

where "n" represents the number of objects to choose from, "r" represents the number of objects to choose without repetition, and "!" denotes the factorial of a number.

Now, let's derive the formula for combinations with repetition from the formula for combinations without repetition.

Consider a scenario where you have "n" distinct objects and you want to choose "r" objects with repetition allowed. This means that you can choose the same object multiple times.

Step 1: Add Repetition Slots To account for repetition, we can imagine adding "r-1" repetition slots or "dummy" objects to the original "n" objects. These repetition slots will serve as placeholders for the repetitions.

For example, let's say you have 3 distinct candies - chocolate, vanilla, and strawberry - and you want to choose 2 candies with repetition allowed. You can represent this scenario as follows, with "_" denoting the repetition slots:

C V S _ _

Step 2: Apply Combinations without Repetition Formula Now, you can apply the formula for combinations without repetition to choose "r" objects from the total number of objects (including the repetition slots) which is "n + (r-1)". The formula becomes:

(n + (r-1))! / (r! * ((n + (r-1)) - r)!)

Step 3: Simplify the Formula Next, you can simplify the formula by canceling out common terms in the numerator and denominator. The formula becomes:

(n + (r-1))! / (r! * (n - 1)!)

Step 4: Rearrange the Formula Finally, you can rearrange the formula to get it into a more familiar form. Using the property (n + k)! = (n + k) * (n + k - 1)!, the formula becomes:

(n + r - 1)! / (r! * (n - 1)!)

which is the formula for combinations with repetition, denoted as n + r - 1 C r.

So, in summary, the formula for combinations with repetition is derived by adding "r-1" repetition slots to the original "n" objects, applying the formula for combinations without repetition, simplifying the formula, and rearranging it to get it into the form n + r - 1 C r.