Counting, Pigeonhole, Permutations, and Combinations

If you can choose from 5 soups and 3 salads, use the addition principle of counting to determine the total number of options

5+3 = 8

If you can choose from 8 soups/salads, 7 mains, and 6 desserts; what is the total number of combos. Use the multiplication principle of counting

8×7×6 different combos

If you can choose 5 soups what are the total combos with soup

5×7×6

What about total combos with salads if there are 3 salads

3×7×6

If there are 14 socks, how many draws to guarantee a match has been found

7+1 = 8

If the first pair of matching socks was found in the last possible drawing necessary, how many draws are needed until another pair is made?

6 draws left 1 par so 6 + 1 = 7

If, after these first two pairs of matching socks were found, the other drawn socks were placed back into the drawer, how many draws are necessary to guarantee a third matching pair to have been found?

There would be 10 socks left so 5 + 1 = 6 draws

Determine how many 8-character passwords are possible under the following conditions. The only characters allowed are the hexadecimal digits, the letters of the English (or, Latin or Roman) Alphabet, and the letters of the Greek Alphabet. 1) The password's characters must differ from each other.

P(87,8) so 87! / (87- 8)!

The password's characters needn't differ from each other

87^8

The password's first two characters must differ from each other while the remaining characters, which needn't differ from each other nor from the first two, must be hexadecimal digits that correspond to decimal prime numbers.

87 × 86 × 6^6

The password's first two characters must differ from each other but not necessarily from the remaining characters, which must also differ from each other while being hexadecimal digits that correspond to decimal prime numbers.

87 × 86 × 6 × 5 × 4 × 3 × 2 × 1