Discrete Mathematics Week10 Combination with Repetition Question4

Combinations with repetition, sometimes called combinations with replacement, involve selecting items from a set where items can be chosen multiple times. This means that in combination with repetition, you can select the same item more than once. Combinations with repetition are used when you want to count the number of ways to choose items without regard to their order and with the possibility of choosing the same item multiple times.

Here's an example to illustrate combinations with repetition:

Suppose you are at an ice cream parlour, and they offer three flavours of ice cream: Chocolate (C), Vanilla (V), and Strawberry (S). You want to create a three-scoop ice cream cone by selecting from these flavours with replacements (meaning you can choose the same flavour multiple times).

The question is: How many different three-scoop ice cream cones can you create using these flavours?

To find the number of combinations with repetition, you can use the formula:

Number of combinations = (n + r - 1) C r

n represents the number of different items to choose from (in this case, the number of ice cream flavours, which is 3: C, V, S).
r represents the number of items you want to choose (in this case, the number of scoops, which is 3).
So, applying the formula:

Number of combinations = (3 + 3 - 1) C 3 = (5) C 3

To calculate (5) C 3, you can use the binomial coefficient formula:

(5) C 3 = 5! / [3!(5 - 3)!] = 5! / (3! * 2!) = (5 * 4 * 3!) / (3! * 2!) = (5 * 4) / (2 * 1) = 10.

There are 10 different three-scoop ice cream cones you can create using Chocolate, Vanilla, and Strawberry flavours with replacement. These combinations include cones with repeated flavours and various arrangements. For example:

CCC (all Chocolate)
VVV (all Vanilla)
SSS (all Strawberry)
CCV (2 Chocolate, 1 Vanilla)
CCS (1 Chocolate, 2 Strawberry)
CVS (1 Chocolate, 1 Vanilla, 1 Strawberry)
VCV (1 Vanilla, 2 Chocolate)
VVS (1 Vanilla, 1 Vanilla, 1 Strawberry)
SSV (1 Strawberry, 2 Vanilla)
SSS (all Strawberry)
These are the 10 possible combinations of three-scoop ice cream cones you can create with the given flavours and repetition allowed.