Permutations, Combinations, Binomial Theorem
1. Simplify
a)
 n  2 !
 n  1!
1
 n  1 n  n  1
 n  2 n  1 n
b)
c)
 n  2 n  1
d)
 n  2
2. There are 3 roads between Winnipeg and Brandon and 5 roads between Brandon and
Roblin. How many different routes are there from Winnipeg to Roblin through Brandon?
4. In how many ways can all the letters of the word T A B L E be arranged if the vowels
and consonants must alternate?
5. Solve for n in the equation:
 n  1!  1
n!
5
6. How many numbers between 99 and 999 are divisible by 5 and have no repetition of digits?
7. Using the letters from the word PORTAGE:
a)
How many 5 letter arrangements are possible?
Express your answer as a whole number.
b)
How many 7 letter arrangements are possible if “P” must be the first letter and
the letters “T” and “E” must be together? Briefly explain your calculations.
8. Solve for n algebraically:
 n 1! 6  n  3!
9. A father wishes to have a picture taken with his 4 sons. In how many different ways can they
line up for the picture if the father must stand in the middle?
10. Kyle has 15 marbles of the same size in his pocket. Four are white, five are red, and the rest
are blue. If he arranges the marbles in a row on his desk, how many permutations can he
possibly make?
a)
15!
12. Solve for n:
b) 4!5!6!
n
c) 3!4!5!6!
d)
15!
4!5!6!
P2  12
13. Using the digits 1, 2, 3, 4, 5, 6, how many 3-digit numbers less than 450 can be formed if the
repetition of digits is not allowed? Briefly explain your calculations.
C
x
14. Solve for x algebraically: n 4 
P
5!
n 4
15. A party of 18 people is divided into 2 different groups consisting of 11 people and 7 people.
The number of different ways this can be done is:
a)
18!
18!

7!
11!
b)
18!
11!7!
c)
18!18!
7!11!
d) 11!7!2!
16. A hockey league has 8 teams. If each team plays every other team only once, how many
games will be played?
a)
8 C2
b)
c) 8 2
8 P2
d) 8!
17. A grad committee of 4 students is to be selected from a class of 24 students. Annie, Betty
and Connie decide that they do not want to be on the committee. How many different
committees are possible? Express your answer in factorial notation.
18. In the binomial expansion  x  2  , the 6th term is 8064x5 . The value for n is:
n
a) 5
c) 10
b) 6
d) 11
19. In the binomial expansion of  a  b  , what is the numerical coefficient of the term
15
containing a 2b13 ?
20. There are 12 terms in the simplified binomial expansion of  x  y  . Find the value of n.
n
4

21. Find the simplified expression for the 6 term in the expansion of:  x3  
x

8
th
10
3

22. In the binomial expansion of  x 2   there is a term that when simplified contains x8 .
x

Find and simplify this term completely.