Download In the questions below consider all bit strings of length 12

1.
The figure shows a 4-block by 5-block grid of streets. Find the number of ways in which
you can go from point A to point B, where at each stage you can only go right or up. (You
are not allowed to go left or down.) For example, one allowable route from A to B is: Right,
Right, Up, Right, Up, Up, Right, Right, Up.
Ans: C(9,4).
2.
How many ways are there for a horse race with five horses to finish if ties are possible?
5=5
1
=4+1
 C(5,1)2!
=3+2
 C(5,2)2!
=3+1+1
 C(5,3)3!
=2+2+1
 C(5,1)  C(4,2)/2 3!
=2+1+1+1
 C(5,2) 4!
= 1 + 1 + 1 + 1 + 1  5!
3. Find the number of permutations of the word COEFFICIENT.
Ans: 11!/(2! 2! 2! 2! 1! 1! 1!).
In the questions 4-7 consider all bit strings of length 10.
4. How many begin with 11 and end with 10? Ans: 26.
5.
6.
7.
How many begin with 11 or end with 10? Ans: 228-26.
How many have exactly four 1s? Ans: C(10,4).
How many contain at least four consecutive 0s?
0 0 0 0 _ _ _ _ _ _ Ans: 26 + 625
10000_____
_10000____
__10000___
___10000__
____10000_
_____10000
8. Suppose  A   4 and  B   10. Find the number of functions f  A  B. Ans: 104.
9. Suppose  A   4 and  B   10. Find the number of 1-1 functions f  A  B. Ans: 10987.
10. Suppose  A   10 and  B   4. Find the number of 1-1 functions f  A  B. Ans: 0.
In the questions 11-18 consider all subsets of S  123…10.
11. How many subsets contain the number 5? Ans: 29.
12. How many subsets contain neither 5 nor 6? Ans: 28.
13. How many subsets contain no odd numbers? Ans: 25.
14. How many subsets contain exactly three elements? Ans: C(10,3).
15. How many subsets contain exactly three elements, one of which is 3? Ans: C(9,2).
16. How many subsets contain exactly three elements, all of them even? Ans: C(5,3).
17. How many subsets contain exactly five elements, the sum of which is even?
O+O+O+O+O = O
O+O+O+O+E = E  C(5,4)C(5,1)
O+O+O+E+E = O
O+O+E+E+E = E  C(5,2)C(5,3)
O+E+E+E+E = O
E+E+E+E+E = E C(5,5)
18. How many subsets contain exactly four elements, the sum of which is even?
O+O+O+O = E C(5,4)
O+O+O+E = O
O+O+E+E = E  (C(5,2))2
O+E+E+E = O
E+E+E+E = E  C(5,4)