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There are two 0's at the end of \(10 !=3,628,800 .\) Using only pencil and paper, determine how many 0's are at the end of the number \(100 !\)

A bag contains 20 identical red balls, 20 identical blue balls, 20 identical green balls, one white ball, and one black ball. You reach in and grab 20 balls. How many different outcomes are possible?

Consider 4-card hands dealt off of a standard 52-card deck. How many hands are there for which all 4 cards are of the same suit or all 4 cards are red? Solution: Let \(A\) be the set of 4 -card hands for which all four cards are of the same suit. Let \(B\) be the set of 4 -card hands for which all four cards are red. Then \(A \cap B\) is the set of 4 -card hands for which the four cards are either all hearts or all diamonds. The answer to our question is \(|A \cup B|=|A|+|B|-|A \cap B|=\) \(4\left(\begin{array}{c}13 \\\ 4\end{array}\right)+\left(\begin{array}{c}26 \\\ 4\end{array}\right)-2\left(\begin{array}{c}13 \\\ 4\end{array}\right)=2\left(\begin{array}{c}13 \\\ 4\end{array}\right)+\left(\begin{array}{c}26 \\\ 4\end{array}\right)=1430+14,950=\mathbf{1 6}, \mathbf{3 8 0}\).

Find how many 9 -digit numbers can be made from the digits 1,2,3,4,5,6,7 , 8,9 if repetition is not allowed and all the odd digits occur first (on the left) followed by all the even digits (i.e., as in 137598264 , but not 123456789 ).

In how many ways can you place 20 identical balls into five different boxes?

A password on a certain site must be five characters long, made from letters of the alphabet, and have at least one upper case letter. How many different passwords are there? What if there must be a mix of upper and lower case?

A coin is tossed 10 times in a row. How many possible sequences of heads and tails are there?

This problem concerns lists made from the symbols \(A, B, C, D, E, F, G, H, I .\) (a) How many length- 5 lists can be made if there is no repetition and the list is in alphabetical order? (Example: BDEFI or \(A B C G H,\) but \(\operatorname{not} B A C G H .\) ) (b) How many length-5 lists can be made if repetition is not allowed and the list is not in alphabetical order?

Compute how many 7 -digit numbers can be made from the digits 1,2,3,4,5,6,7 if there is no repetition and the odd digits must appear in an unbroken sequence.

A new car comes in a choice of five colors, three engine sizes and two transmissions. How many different combinations are there?