Help with factorials, combinations and permutations

1. B. Heard
(Not to be posted, used, etc. without my
permission, students may download one
copy for personal use)

2.  Inany statistics class, you will come across
factorials, permutations and combinations
 You can use Minitab, your calculator, etc. for
these types of calculations
 I am going to show you how to use Excel (because it is
usually something everyone has on their computer)

3. Factorials

4. The factorial symbol “!” (an exclamation
point) simply means the product of decreasing
positive whole numbers.
For example, 4! = 4x3x2x1 = 24
Always remember that 0! = 1.

5. A little practice
 5! = 5x4x3x2x1 = 120
 3! + 4! = (3x2x1) + (4x3x2x1) = 6 + 24 = 30
 (7-3)! = 4! = 4x3x2x1 = 24
 7! – 3! = 7x6x5x4x3x2x1 – 3x2x1 = 5040 – 6 = 5034
 4!/0! = (4x3x2x1)/1 = 24
 5(0!) = 5(1) = 5

6.  You can easily do these in Excel by using
“=FACT(#)” (not including the quotation
marks and putting the number in for the #
sign)
 In other words, pick a blank cell in Excel and
type
=FACT(7)
And hit the enter key
You will see your result of 5040 which is 7!

7. Hit Enter and you get
5040

8.  What about a big number?
 Enter “=FACT(25)” and you will get
 Thisis so large that exponential notation has
to be used
 1.55112E+25 in Excel just means 1.55112x 1025 or
 15511200000000000000000000.0
I moved the decimal 25 places to the right

9. Permutations

10. Permutation Requirements:
1. There are n different items available. (This rule does not
apply if some of the items are identical to others.)
2. We select r of the n items (without replacement).
3. We consider rearrangements of the same items to be different
sequences. (The permutation of ABC is different from CBA and
is counted separately.) This just means “order matters” – I will
give you examples
If the preceding requirements are satisfied, the number of
permutations (or sequences) of r items selected from n available
items (without replacement) is
n Pr = n!
(n - r)!

11.  There are 10 finalists in a local radio
station’s contest. From the 10 finalists, 3
winners will be selected to win one of three
prizes. The prizes are $10,000 , dinner for
two at a local restaurant and a t-shirt from
the radio station.
 How many ways can the 3 winners be
selected from the 10 finalists?

12.  First ask yourself, “Does order matter?”
 Yes, had you rather have $10,000 or the t-shirt? (the
prizes are distinct)
 So this is a permutation
10 P3 10!
=
(10 - 3)!

13.  You can easily do these in Excel by using
“=PERMUT(n,p)” (not including the
quotation marks and putting the number in
for n and p and don’t forget the comma in
between)
 In other words, pick a blank cell in Excel and
type
=PERMUT(10,3)
And hit the enter key
You will see your result of 720

14. Type =PERMUT(10,3) and
hit Enter you get 720

15.  Another Example
 There are 12 students in a classroom and a
committee of 4 is to be chosen. The
committee has a President, VP, Secretary and
Treasurer
 How many ways can the 4 be selected from
the 12 students?

16.  First ask yourself, “Does order matter?”
 Yes, there are distinct positions.
 So this is a permutation
12 P4 12!
=
(12 - 4)!

17.  Picka blank cell in Excel and type
=PERMUT(12,4)
And hit the enter key
You will see your result of 11880

18. Combinations

19. Combination Requirements:
1. There are n different items available.
2. We select r of the n items (without replacement).
3. We consider rearrangements of the same items to be the same.
(The combination of ABC is the same as CBA.) This is saying
“order does not matter.”
If the preceding requirements are satisfied, the number of
combinations of r items selected from n different items is
n!
nCr = (n - r )! r!

20.  There are 10 finalists in a local radio
station’s contest. From the 10 finalists, 3
winners will be selected to win $1,000.
 How many ways can the 3 winners be
selected from the 10 finalists?

21.  First ask yourself, “Does order matter?”
 No, all three get the same prize.
 So this is a combination.
10!
10C3 = (10 - 3 )! 3!

22.  You can easily do these in Excel by using
“=COMBIN(n,p)” (not including the quotation
marks and putting the number in for n and p and
don’t forget the comma in between)
 In other words, pick a blank cell in Excel and
type
=COMBIN(10,3)
And hit the enter key
You will see your result of 120
Note that the number of combinations is less than
permutations because we don’t have to consider
order

23.  Another Example
 There are 12 students in a classroom and a
committee of 4 is to be chosen.
 How many ways can the 4 be selected from
the 12 students?

24.  First ask yourself, “Does order matter?”
 No, there are no distinct positions
mentioned, only a committee of 4.
 So this is a combination of 12C4 .
 Just use “=COMBIN(12,4)” in an Excel cell and hit
the Enter key to get your answer of 495

25.  What about those funky questions about how
many distinct words or orderings you can
make with the letters in the word “Colorado”
or “Mississippi” etc.?
 These are “distinct permutation” problems
and can be easily explained with examples.
 In these problems you generally have some
duplicates you have to deal with.

26. Permutations with we have identical items Summary:
1. There are n items available, and some items are identical to
others.
2. We select all of the n items (without replacement).
3. We consider rearrangements of distinct items to be different
sequences.
If the preceding requirements are satisfied, and if there are n1
alike, n2 alike, . . . nk alike, the number of permutations (or
sequences) of all items selected without replacement is
n!
n1! . n2! .. . . . . . .
n!

27.  How many different orderings can be made from
the letters in the word “book?”
 Ask yourself, how many total letters are there?
 Then ask yourself, how many of each different
letter?
4 total letters 4! 4x3x2x1
1 “b” =
1!x2!x1 1x2x1x1
2 “o’s”
1 “k” !
=12 (answer)

28.  How many different orderings can be made from
the letters in the word “Mississippi?”
 Ask yourself, how many total letters are there?
 Then ask yourself, how many of each different
letter?
 There are 11 total letters (if I spelled it right).
 1 M, 4 I’s, 4 S’s and 2 P’s
11!
1!x4!x4!x2!

29.  For this calculation, use your calculator or
Excel, or do it by hand.
 You can do it in Excel, but you have to use
parentheses
 In Excel you would type
=FACT(11)/(FACT(1)*FACT(4)*FACT(4)*FACT(2))
 You have to watch your parentheses and not
forget about your *’s for multiplication
 The answer is 34650 (I checked by hand also)
 The 1!’s in the denominator are unnecessary
because 1! Is one, but I say put them in because
it helps you understand!

30.  Similar problem
 If there are 9 flags on a pole including 3 identical
red flags, 2 identical blue flags and 4 identical
green flags, how many different ways or patterns
can be made?
9! = 1260 different ways
3!x2!x4!

31.  Good luck, I hope this helps.
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