how many ways???

Question

can you make change for 50 cent

Answer 1

There are 50 ways, if you include 1 50 cent piece as one way.
(49 ways otherwise)
This problem is equivalent to finding the number of partitions of 50 with summands restricted to 1,5,10, 25 and 50.
The answer is then the coefficient of x^50 in the expansion of the product P1P2P3P4P5 where the P's are the polynomials:

P1 = 1 + x + x^2 + x^3 + x^4 + ... + x^50
P2 = 1 + x^5 + x^10 + x^15 + ... + x^50
P3 = 1 + x^10 + x^20 + x^30 + ... + x^50
P4 = 1 + x^25 + x^50
P5 = 1 + x^50
All calculations will be made up to x^50

P4P5 = 1 + x^25 + 2x^50

P3P4P5 = 1 + x^10 + x^20 + x^25 + x^30 + x^35 + x^40 + x^45 + 3x^50 ...

P2P3P4P5 = 1 + x^5 + 2x^10 + 2x^15 + 3x^20 + 4x^25 + 5x^30 + 6x^35 + 7x^40 + 8x^45 + 11x^50

The final multiplication need not be done in detail, since we are concerned only with the coefficient of x^50. We notice that each term in the polynomial product P2P3P4P5 enters exactly once in contributing to the coefficient of x^50 in the product of P1 and P2P3P4P5. It follows that this coefficient can be calculated simply by adding all the coefficients of P2P3P4P5 (including the constant term 1):
1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 11
This sum is 50, so there are 50 ways to make 50 cents with pennies, nickels, dimes, quarters and halfs.

That was the pure mathematics solution. Here is the list:
Note that the number of pennies is defined by the number of the other coins, so it amounts to showing the number of ways you can make less than or equal to 50 cents with the other coins. Then the ways are (nickels,dimes,quarters):

0,0,0
0,0,1
0,0,2
0,1,0
0,1,1
0,2,0
0,2,1
0,3,0
0,4,0
0,5,0
1,0,0
1,0,1
1,1,0
1,1,1
1,2,0
1,2,1
1,3,0
1,4,0
2,0,0
2,0,1
2,1,0
2,1,1
2,2,0
2,3,0
2,4,0
3,0,0
3,0,1
3,1,0
3,1,1
3,2,0
3,3,0
4,0,0
4,0,1
4,1,0
4,2,0
4,3,0
5,0,0
5,0,1
5,1,0
5,2,0
6,0,0
6,1,0
6,2,0
7,0,0
7,1,0
8,0,0
8,1,0
9,0,0
10,0,0
and
1 50 cent piece

Answer 2

Look at ways to get less than 50 cents or less with quarters and dimes only:


a) 2 quarters, no dimes
b) 1 quarter, no dimes
c) 1 quarter, 1 dime
d) 1 quarter, 2 dimes
e) no quarters, no dimes
f) no quarters, 1 dime
g) no quarters, 2 dimes
h) no quarters, 3 dimes
i) no quarters, 4 dimes
j) no quarters, 5 dimes

No, in each case you need to make up the rest of the money in pennies and nickels. If you want to make 5 cents from pennies and nickels, there is two ways. If you want to make 10 cents from pennies and nickels, there are three ways. 15 cents, four ways, etc.

(a) 1 way
(b) 6 ways
(c) 4 ways
(d) 2 ways
(e) 11 ways
(f) 9 ways
(g) 7 ways
(h) 5 ways
(i) 3 ways
(j) 1 way

That comes to 49 different ways.

Answer 3

Lots!

2 quarters
5 dimes
10 nickels
50 pennies

plus many combinations of the above....

1 quarter + 5 nickels
1 quarter + 2 dimes + 1 nickel
1 quarter + 1 dime + 3 nickels
1 quarter + 2 dimes + 5 pennies

ad nauseum.

I will leave it to you to find all the possible combinations! (Or some other answerer)

.

Answer 4

A lot. You can use 5c, 10c, 20c. Try to use probability to solve this question by using the method "Combination"