Question

Peter planned to have a wedding party at his favorite restaurant. He would order 48 main dishes and 48 desserts for the guests.If 3 types of main dishes and 4 types of desserts were provided, how many different combinations could Peter order (he could choose same types of dishes more than one time)? Explain why the answer is 25510625.

Answer 1

To solve this problem we will use Combinations with repetitions

For main dishes number of available options = 3

For desserts number of available options = 4

Number of main dishes Peter want to order = 48

Number of desserts Peter want to order = 48

Combinations with repetitions formula :

it is denoted by while normal combinations is denoted by C

Formula =

Now,

Number of ways 48 main dishes can be chosen from 3 main dishes with repetitions :-

here n= 3 and p = 48

so, Number of ways 48 main dishes can be chosen from 3 main dishes with repetitions = (3,48) = C(3+48-1,48)

Here, (3,48) =C(3+48-1,48) = C(50,48) = =   = 1225

we can get the value of (3,48) from Combinations with repetitions formula mentioned above or we can use C(50,48) which is equal to (3,48) and to get the value of C(50,48) we can use buit in Combination function in calculator or combination formula mathematically just as we normally calculate Combinations.

Number of ways 48 main dishes can be chosen from 3 main dishes with repetitions = 1225

Next we will compute

Number of ways 48 desserts can be chosen from 4 desserts with repetitions :-

here n= 4 and p = 48

Number of ways 48 desserts can be chosen from 4 desserts with repetitions = (4,48) = C(4+48-1,48) = C(51,48)

C(51,48) = 20825

Number of ways 48 desserts can be chosen from 4 desserts with repetitions = 20825

Now to get how many different combinations could Peter order we multiply number of ways 48 main dishes can be chosen from 3 main dishes with repetitions and number of ways 48 desserts can be chosen from 4 desserts with repetitions so that we can get all the different combination Peter can order

Number of different combination Peter could order = (Number of ways 48 main dishes can be chosen from 3 main dishes with repetitions Number of ways 48 desserts can be chosen from 4 desserts with repetitions)

Number of different combination Peter could order = (1225 20825) = 25510625

Answer: 25510625 different combinations could Peter order