Question

A large pile of coins consists of pennies, nickels, dimes, and quarters.

a. how many different collections of 30 coins can be chosen if there are at least 30 of each kind of coin?

Answer 1

There are 5456 different collections of 30 coins that can be chosen from pennies, nickels, dimes, and quarters, assuming there is an unlimited supply of each kind of coin.

Step-by-step explanation:

The question is asking to find out how many different collections of 30 coins can be made from an unlimited supply of pennies, nickels, dimes, and quarters. We are dealing with a combination problem in the field of combinatorics, which is a part of mathematics dealing with countable discrete structures.

To find the answer, we can use the formula for the number of combinations of n objects taken r at a time with repetition allowed (also known as the stars and bars method):

C(n + r - 1, r) = C(30 + 4 - 1, 30) = C(33, 30)

Where C(n, r) is the binomial coefficient which can be calculated as:

C(n, r) = n! / (r! * (n - r)!)

Using this formula, we get:

C(33, 30) = 33! / (30! * (33 - 30)!)
= 33 * 32 * 31 / (3 * 2 * 1)
= 5456

Therefore, there are 5456 different collections of 30 coins that can be chosen.

Answer 2

You can think of 4 distinct piles, one for each coin denomination, into which you can put any number of coins from 0 to 30 so that the total is 30.