Practice for the first midterm

Determine the largest power of 10 that is a factor of the following numbers (equivalently, the number of terminal 0s, using ordinary base 10 representation):

(a) 50! : __
(b) 1000! : ___

(a)  

How many 6 digit numbers have both of the following properties?

(a) The digits are distinct.

(b) The digits 2 and 7 do not occur.
Answer: (a)  

In how many ways can four men and eight women be seated at a round table if there are to be two women between consecutive men around the table? Answer: (a)  

In how many ways can six indistinguishable rooks be placed on a 6-by-6 board so that no two rooks can attack one another?

___

In how many ways if there are two red and four blue rooks? _____

(a)  

What is the number of necklaces that can be made from 20 beads, each of a different color?

(a)  

What is the number of ways to order the 26 letters

of the alphabet so that no two of the vowels a, e, i, o and u occur consecutively?

There are 20 identical sticks lined up in a row occupying 20 distinct places as follows:
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Six of them are to be chosen.

(a) How many choices are there?

(b) How many choices are there if no two of the chosen sticks can be consecutive?

(c) How many choices are there if there must be at least two sticks between each pair of chosen sticks?

In how many ways can 12 indistinguishable apples and 1 orange be distributed among three children in such a way that each child gets at least one piece of fruit?

(a)