1.Using the principle of inclusion-exclusion find the numbers of integers between 1 and 1000 (inclusive)that are divisible by at least one of 2,3,5,or 7?

2.A drug store sells gum, candy, and playing cards. 15 teenagers are in the store...the clerk notes after the teenagers leave that 10 bought gum, 7 bought candy, and 12 purchased playing cards. 5 bought gum and candy,6 bought gum and playing cards and 2 purchased candy and playing cards...Why is the clerk so upset?

1. To find the numbers of integers between 1 and 1000 that are divisible by at least one of 2, 3, 5, or 7, we can use the principle of inclusion-exclusion.

Step 1: Count the numbers divisible by 2.
There are 1000/2 = 500 numbers divisible by 2 between 1 and 1000.

Step 2: Count the numbers divisible by 3.
There are 1000/3 = 333 numbers divisible by 3 between 1 and 1000.

Step 3: Count the numbers divisible by 5.
There are 1000/5 = 200 numbers divisible by 5 between 1 and 1000.

Step 4: Count the numbers divisible by 7.
There are 1000/7 = 142 numbers divisible by 7 between 1 and 1000.

Step 5: Count the numbers divisible by both 2 and 3.
There are 1000/(2*3) = 166 numbers divisible by both 2 and 3 between 1 and 1000.

Step 6: Count the numbers divisible by both 2 and 5.
There are 1000/(2*5) = 100 numbers divisible by both 2 and 5 between 1 and 1000.

Step 7: Count the numbers divisible by both 2 and 7.
There are 1000/(2*7) = 71 numbers divisible by both 2 and 7 between 1 and 1000.

Step 8: Count the numbers divisible by both 3 and 5.
There are 1000/(3*5) = 66 numbers divisible by both 3 and 5 between 1 and 1000.

Step 9: Count the numbers divisible by both 3 and 7.
There are 1000/(3*7) = 47 numbers divisible by both 3 and 7 between 1 and 1000.

Step 10: Count the numbers divisible by both 5 and 7.
There are 1000/(5*7) = 28 numbers divisible by both 5 and 7 between 1 and 1000.

Step 11: Count the numbers divisible by all of 2, 3, 5, and 7.
There are 1000/(2*3*5*7) = 4 numbers divisible by all of 2, 3, 5, and 7 between 1 and 1000.

Now, using the principle of inclusion-exclusion, we can find the numbers divisible by at least one of 2, 3, 5, or 7:
Total = numbers divisible by 2 + numbers divisible by 3 + numbers divisible by 5 + numbers divisible by 7 - numbers divisible by both 2 and 3 - numbers divisible by both 2 and 5 - numbers divisible by both 2 and 7 - numbers divisible by both 3 and 5 - numbers divisible by both 3 and 7 - numbers divisible by both 5 and 7 + numbers divisible by all of 2, 3, 5, and 7.

Total = 500 + 333 + 200 + 142 - 166 - 100 - 71 - 66 - 47 - 28 + 4
Total = 1137

Therefore, there are 1137 integers between 1 and 1000 (inclusive) that are divisible by at least one of 2, 3, 5, or 7.

2. The clerk is upset because the numbers provided by the teenagers do not match up. According to the given information, 10 teenagers bought gum, 7 bought candy, and 12 purchased playing cards. However, there are discrepancies between overlapping purchases. For example, 5 teenagers bought both gum and candy, 6 bought both gum and playing cards, and 2 bought both candy and playing cards. The numbers don't add up correctly, leading to confusion and frustration for the clerk.