SOLUTION: All the five units of money in a womans bag are in different denominations. How many different sums of money can she form?

For the smallest denomination of money, there are two choices:
1. Include it in the sum.
2. Do not include it in the sum.

That's 2 choices for the smallest denomination in making a
sum of money.

For each of those 2 choices for the smallest denomination of money, 
there are 2 choices to make for the next to smallest denomination
of money:
1. Include it in the sum.
2. Do not include it in the sum.

So that's 2x2 or 4 choices to make for the smallest and next to
smallest denominations in making the sum of money.

For each of those 4 choices for the smallest and next to smallest
denominations of money,  there are 2 choices to make for the middle
denomination of money:
1. Include it in the sum.
2. Do not include it in the sum.

So that's 2x2x2 or 4x2 or 8 choices to make for the smallest two 
and the middle-sized denominations in making the sum of money.

For each of those 8 choices for the smallest, next to smallest, and
middle-sized denominations of money,  there are 2 choices to make 
for the next to largest denomination of money:
1. Include it in the sum.
2. Do not include it in the sum.

So that's 2x2x2x2 or 8x2 or 16 choices to make for the smallest two, 
the middle-sized, and the next to largest denominations in making 
the sum of money.

For each of those 16 choices for the smallest, next to smallest,
middle-sized, and next to largest denominations, there are 2 choices 
to make for the largest denomination of money:
1. Include it in the sum.
2. Do not include it in the sum.

So that's 2x2x2x2x2 or 16x2 or 32 choices to make in making 
the sum of money.

Answer: 2x2x2x2x2 = 25 = 32, including the case
where she does not include any of them and gets a sum of $0.
If you don't want to count that case, then subtract 1 and
the answer is only 31.

Checking: suppose she has one each of $1, $5, $10, $50, and $100 bills.

Then she can make

 1.  0×$1 + 0×$5 + 0×$10 + 0×$50 + 0×$100 = $0
 2.  0×$1 + 0×$5 + 0×$10 + 0×$50 + 1×$100 = $100
 3.  0×$1 + 0×$5 + 0×$10 + 1×$50 + 0×$100 = $50
 4.  0×$1 + 0×$5 + 0×$10 + 1×$50 + 1×$100 = $150
 5.  0×$1 + 0×$5 + 1×$10 + 0×$50 + 0×$100 = $10
 6.  0×$1 + 0×$5 + 1×$10 + 0×$50 + 1×$100 = $110
 7.  0×$1 + 0×$5 + 1×$10 + 1×$50 + 0×$100 = $60
 8.  0×$1 + 0×$5 + 1×$10 + 1×$50 + 1×$100 = $160
 9.  0×$1 + 1×$5 + 0×$10 + 0×$50 + 0×$100 = $5
10.  0×$1 + 1×$5 + 0×$10 + 0×$50 + 1×$100 = $105
11.  0×$1 + 1×$5 + 0×$10 + 1×$50 + 0×$100 = $55
12.  0×$1 + 1×$5 + 0×$10 + 1×$50 + 1×$100 = $155
13.  0×$1 + 1×$5 + 1×$10 + 0×$50 + 0×$100 = $15
14.  0×$1 + 1×$5 + 1×$10 + 0×$50 + 1×$100 = $115
15.  0×$1 + 1×$5 + 1×$10 + 1×$50 + 0×$100 = $65
16.  0×$1 + 1×$5 + 1×$10 + 1×$50 + 1×$100 = $165
17.  1×$1 + 0×$5 + 0×$10 + 0×$50 + 0×$100 = $1
18.  1×$1 + 0×$5 + 0×$10 + 0×$50 + 1×$100 = $101
19.  1×$1 + 0×$5 + 0×$10 + 1×$50 + 0×$100 = $51
20.  1×$1 + 0×$5 + 0×$10 + 1×$50 + 1×$100 = $151
21.  1×$1 + 0×$5 + 1×$10 + 0×$50 + 0×$100 = $11
22.  1×$1 + 0×$5 + 1×$10 + 0×$50 + 1×$100 = $111
23.  1×$1 + 0×$5 + 1×$10 + 1×$50 + 0×$100 = $61
24.  1×$1 + 0×$5 + 1×$10 + 1×$50 + 1×$100 = $161
25.  1×$1 + 1×$5 + 0×$10 + 0×$50 + 0×$100 = $6
26.  1×$1 + 1×$5 + 0×$10 + 0×$50 + 1×$100 = $106
27.  1×$1 + 1×$5 + 0×$10 + 1×$50 + 0×$100 = $56
28.  1×$1 + 1×$5 + 0×$10 + 1×$50 + 1×$100 = $156
29.  1×$1 + 1×$5 + 1×$10 + 0×$50 + 0×$100 = $16
30.  1×$1 + 1×$5 + 1×$10 + 0×$50 + 1×$100 = $116
31.  1×$1 + 1×$5 + 1×$10 + 1×$50 + 0×$100 = $66
32.  1×$1 + 1×$5 + 1×$10 + 1×$50 + 1×$100 = $166

The list includes the very first case where
the sum is $0 where she doesn't choose any 
of the bills.  If you don't want to count that
case then the answer is 31.

Edwin