$\begingroup$So I thought for the first day we have $4 \cdot 5$ outfits and for the second day $3 \cdot 4$ outfits for a total of $20 + 12 = 32$. Is this reasoning correct?
$\endgroup$41 Answer
$\begingroup$Method 1: We consider cases:
- A different pair of pants is worn each day and a different shirt is worn each day. The first day, there are five ways to select a pair of pants and four ways to select a shirt, giving $5 \cdot 4 = 20$ possible outfits. After those clothes are consigned to the laundry, we have four ways of selecting a shirt and three ways of selecting a pair of pants on the second day, enabling us to select $4 \cdot 3 = 12$ outfits. Since there are $12$ choices on the second day for each of the $20$ choices on the first day, there are $20 \cdot 12 = 240$ ways to select the outfits in this case.
- The same pants are worn both days with two different shirts. There are four ways to select the pants, five ways to select the first shirt, and four ways to select the second shirt. Thus, $4 \cdot 5 \cdot 4 = 80$ such outfits can be selected.
- The same shirt is worn both days with two different pairs of pants. There are five ways to select the shirt, four ways to select the first pair of pants, and three ways to select the second shirt. Thus, $5 \cdot 4 \cdot 3 = 60$ such outfits can be selected.
In total, this yields $240 + 80 + 60 = 380$ different possible outfits.
Method 2: We subtract the number of ways of selecting the same outfit from the total number of possible outfits. If the same clothes can be worn each day, there are $20$ ways to select an outfit each day, yielding $20 \cdot 20 = 400$ possible sequences of outfits over the two days. However, $20$ of these sequences require wearing the same outfit on both days. Thus, the total number of different outfits that can be worn over the two days is $400 - 20 = 380$.
$\endgroup$3
