Give an example of a counting problem where 2^5 - 1 is the solution. Explain why you must subtract 1.

Suppose a problem of the form:

A person can choose:

1 out of 2 pants (red or blue)

1 out of 2 shirts (red or blue)

1 out of 2 pairs of shoes (red or blue)

1 out of 2 socks (red or blue)

1 out of 2 hats (red or blue)

Ok,

We have 5 selections with 2 options each, so the number of combinations is:

2^5

Now let's add another restriction, the person can not wear all red clothes, the person must have at least one blue one.

So we removed the combination where the person selects all red clothes, which means that we removed one combination.

So now, the total number of possible combinations is the number that we got before minus one:

2^5 - 1

Notice that subtracting the one is necessary, we first use the math that we know to find the total number of combinations and then we look at the restrictions to remove the restricted combination.

Then the complete problem is:

A person can choose:

1 out of 2 pants (red or blue)

1 out of 2 shirts (red or blue)

1 out of 2 pairs of shoes (red or blue)

1 out of 2 socks (red or blue)

1 out of 2 hats (red or blue)

And the person can not wear all red clothes, the person must have at least one blue one.

How many different combinations there are?