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5.3 Problem Solving With Combinations (Part 1)
Overview
This chapter will look at situations where you want to know the total number of possible combinations of any size that you could choose from a given number of items, where some may be identical. Diction
- Total set of items is referred to as ‘n’.
- Items being chosen from a set are referred to as ‘r’.
- A null set of items is a set that has no elements.(ie. 6C0=1)
-Retain the null set when the question implies that no object at all can also be selected or states: up to, maximum of…· When using combinations you can:
- Remove the null set when the question implies that some objects (at least 1) have to be selected.
1. Choose a specific number of objects ‘r’ from the set ‘n’.
2. Choose a different ‘r’ in each case.-Use cases in questions with terminology like:-Either/or
- At least 1,
-1 or more
-2 or more
- Minimum of
- Maximum of
- Up to
- The terminology has a great importance in getting an answer correct or incorrect. Pay attention to it.
Formula
- Combination notation nCr--> = n! (see example 2 and flowchart)
2. Combinations in which some items are identical
§ (p+1) (q+1) (s+1) …-1 [to remove the null set] (see example 3)
§ (p+1) (q+1) (s+1) … [to include the null set] (see example 3)
3. Not choosing exactly ‘r’ objects, all distinct objects and including the null set: (2^n)
4. Not choosing exactly ‘r’ objects, all distinct objects and excluding the null set: (2^n) -1
5. Not choosing exactly ‘r’ objects, all distinct objects and at least 2 objects are selected: (2^n)- nCr = (2^n) – (nCr0+nCr1+…+ nCri)
NOTE: 2^n = 2 to the power of n
Combinations Lesson (Flowchart)
Examples
Example 1. Using exactly r objects, with identical items. Problem - If you are dealing from a standard deck of 52 cards, how many 3 card hands could have at least one spade? Solution - recall from the flowchart, we use Cases.
Solution-
Step 1- Write out the cases.
Case 1: Only one spade from 13 spades, two other cards from remaining 39 cards.
13C1 x 39C2 = 13 x 741 = 9 633
Case 2: Two spades from 13 spades, one other card from remaining 39 cards. 13C2 x 39C1 = 78 x 39 = 3042
Case 3: Three spades from 13 spades, zero other cards from remaining 39 cards. 13C3 x 39C0 = 286 x 1 = 286
9 633 + 3042 + 286 = 12 961
Therefore there are 12 961 combinations possible of dealing a 3 card hand from 52 cards and deal out at least 1 spade.
**this can be done with a 5 card hand. Just use the 3 card hand as a simpler model.**
Example 2. Using exactly ‘r’ objects, with no identical objects.
Problem - In how many ways can a 4 colors be drawn from a pack of 24 pencil crayons?
Solution - Recall from the flow chart, we use the formula nCr = n!
(n-r)!r!
n=24; r=4
Way1-performed using a calculator Way 2-calculated by hand nCr = n!
(n-r)!r!= 24C4 = 24!
(24-4)! 4!= 10 626 ways = 24!
20!4!= 10 626 ways
Either way there are 10 626 ways of drawing 4 pencils from a pack of 24 pencil crayons.
Example 3. Not using exactly ‘r’ objects, but with some identical objects.
Problem - How many sums of money can i make with 1 penny, 2 nickles and 1 dime .
Solution - Recall from flow chart we use the formula
(p+1)(q+1)(s+1) … or
(p+1)(q+1)(s+1) …-1
(p+1)(q+1)(s+1) …-1
Step 1 -
- Let p represent the number of pennies
- Let q represent the number of nickles
- Let s represent the number of dimes
Step 2 -
Lets substitute values into (p+1)*(q+1)*(s+1)
Step 3 -
= (1+1)*(2+1)*(1+1)
= (2)*(3)*(2)
=12
Since a combination of money is required, the null set cannot be used. [ (p+1)(q+1)(s+1) …-1 ]
Step 4 -
=12-1
=11
Therefore 11 sums of money can be made.
Example 4. Not using exactly ‘r’ objects and with no identical objects.
Problem - The school band consists of 11 members. In how many ways can a committee of at least one member be formed from the school band?
Solution - Recall from the flow chart, the formula (2^n)-1
n=11
2^11-1
= (2x2x2x2x2x2x2x2x2x2x2) -1
=2048 - 1
= 2047
Therefore 2047 committees can be formed from the group of 11 band members with at least one member on a committee.
References
Canton, Barbara J., Erdman, Wayne, Irvine, Jeff, Lim, Louis, Mclaren, Fran, Meisel,Roland W., Miller, David T., Speijer, Jacob. (2002). Mathematics of Data
Management. Toronto: McGraw - Hill.
(2006, July 7). Combinations Flowchart. Retrieved 2006, Nov 18 from
Latest page update: made by Tyndale11
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| Started By | Thread Subject | Replies | Last Post | |
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| Anonymous | Where's Part 2?? | 0 | Mar 22 2008, 7:35 PM EDT by Anonymous | |
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Thread started: Mar 22 2008, 7:35 PM EDT
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Hi! thanx a lot for this.. i was just wondering.. where's Part 2??
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| n_amodan | Suggestion | 1 | Jul 22 2007, 5:52 PM EDT by Anonymous | |
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Thread started: Jan 9 2007, 7:56 PM EST
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Overall your page is really well done, but some of your equations don't look so neat and kind of cluttered. If you made them using Microsoft Equation Editor your equations would have been displayed more neatly and would have been easier to understand.
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| Mr._D'Onofrio | it seems your changes are drastic | 0 | Jun 5 2007, 12:07 AM EDT by Mr._D'Onofrio | |
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Thread started: Jun 5 2007, 12:07 AM EDT
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You were instructed on a handout to inform your teacher if you were going to make drastic changes. Have you informed Ms. Richardson of your changes?
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| Anonymous | comment | 0 | May 26 2007, 5:06 PM EDT by Anonymous | |
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Thread started: May 26 2007, 5:06 PM EDT
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i agree with the first comment. doesn't make sense to a 10th grader.
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| mrunmai | comment | 2 | Nov 27 2006, 7:16 PM EST by mrunmai | |
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Thread started: Nov 26 2006, 11:34 AM EST
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there are some positioning problems and the flowchart can be made bigger
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