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Truth Tables of Seven Teachers and Three Children

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1) How would you disprove this statement? [P and Q] => [P and R]

2) How many multiples of 4 are there in {n; 37< n <1001} ?

3) Seven teachers and three children are to stand in line for a photo session.
a) How many arrangements are there?
b) How many arrangements are there if only any 4 teachers and no children are in the picture?
c) How many arrangements are there if they all come in the picture but with teachers at both ends?
d) How many arrangements are there if two teachers at the end but no two children together?

4) A dog, a cat, a boy and a girl must take the corners of an n-sided polygon. How many arrangements are there if the pets can not occupy contiguous corners? (It is understood that no corner can be occupied by more than one person/animal)
How many if the pets must be next to each other?

5) How many subsets of {1, 2, ? , 10} contain the set {8, 9 , 10} as subset?

6) Express 2 × 4 ×...×18 × 20 by using 2 and the factorial.

7) Social Security numbers (SSN) may start with zero; they have nine digits.
How many SSN are there whose last for digits are different?
How many SSN are there if some of the last four digits coincide?

8) Prove that if 6 divides n + 1, then n^2 + 11 is a multiple of 12.

9) How many numbers are there in {1, 2, . . . . , 1000} which have at least one digit equal to 3?

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The solution discusses truth tables of seven teachers and three children.

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Note that problem 3 is ambiguous. One can solve it if the order of people in the picture ...

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