Download Ch. 11 Review notes

Chapter 11: Permutations, Combinations and Binomial Theorem
The Fundamental Counting Principle
Consider a task made up of several stages. If the number of choices for the first stage
is a, the number of choices for the second stage is b, the number of choices for the third
stage is c, etc., then the number of ways in which a task can be completed is a x b x c x
….. This is called the fundamental counting principle.
Another thing that you will need to consider is the idea of and vs. or. If the problem has
“and” in it, that means you are to multiply. If the problem has “or” or “at least” or “at
most”, it is talking about multiple situations and you need to add.
Factorial Notation
In general n!=n(n-1)(n-2)(n-3)….(3)(2)(1), where n W
On Calc: Math – PRB – 4 – !
Permutations
An arrangement of a set of objects in which the order of the objects is important is
called a permutation.
The number of permutations of “n” different objects taken “r” at a time is:
n!
n Pr  (n  r )!
You can also use the n Pr key on your calculator.
0! Is defined to have a value of 1
Permutations With Restrictions And Repetition
In many problems restrictions are placed on the order in which objects are arranged. In
this type of situation deal with the restrictions first.
Permutations with Repetitions
The number of permutations of n objects, where a are the same
of one type, b are the same of another type, and c are the same
of yet another type, can be represented by the expression below
n!
a !b !c !
**basically remember to divide out the repeats in factorial notation**
One thing that you may see on the diploma are what we call pathway questions. They
can be done by using repeated permutations or drawing.
Combinations
A combination is a selection of a group of objects, taken from a larger group for which
the kind of objects selected is important, but not the order in which they are selected.
There are several ways to find the number of possible combination. One is to use
reasoning. Use the fundamental counting principle and divide by the number of ways
that the object can be arranged among themselves.
Formula:
The number of combinations of “n” items take “r” at a time is:
n!
n Cr   n  r !r !
The n Cr key on the calculator can be used to evaluate combinations:
Math  PRB  3  n Cr
 n
In some texts n Cr is written as  
r 
Pascal’s Triangle: Each entry is found by adding the two numbers above it.
Pascal’s Triangle can also be represented using combinations:
1
2
C0
0
C0
C0
1
2
C1
C1
2
C2
C0 3 C1 3 C2 3 C3
4 C1
4 C0
4 C2
4 C3
4 C4
3
Notice these results from Pascal’s Triangle:
 the sum of the numbers in the kth row of Pascal’s triangle is 2k 1

the sum of the coefficients in the expansion of  x  y  is 2 n

n
n C0  n C1  n C2  ...  n Cn1  n Cn  2
n
2
Binomial Theorem
 x  y
n
 n C0 x n  n C1 x n 1 y  n C2 x n  2 y 2  ...  n Ck x n k y k  ...  n Cn y n , where n  I , n  0
All binomial expressions will be written in descending order of the exponent of the first
term in the binomial. The following are some important observations about the
n
expansion of  x  y  , where x and y represent the terms of the binomial and nεN:


The expansion contains n + 1 terms
The sum of the exponents in any term of the expansion is n.
General Term of the Expansion of ( x  y)n
The term n Ck x n  k y k is called the general term of the expansion.
It is the  k  1 term in the expansion (not term k)
th
t  n C x n k y k
t 1
k
3