NEL 4.7 Solving Counting Problems110

YOU WILL NEED• calculator

Example 1

How many sequences can be formed using each of the digits 0 through 9 exactly once, if• the final digit in the sequence is 0,• a multiple of 4 is the first digit in the sequence, and• another multiple of 4 is either 5th, 6th, or 7th in the sequence?

Solution

Step 1. Order is important, so I knew this problem involves permutations. I also realized

that the problem involves both “and” and “or,” so I would have to consider multiple

cases but also multiple stages.

Step 2. I decided to use the third condition to determine three cases.

Case 1: multiples of 4 in 1st and 5th positions with 0 as the final digit

Case 2: multiples of 4 in 1st and 6th positions with 0 as the final digit

Case 3: multiples of 4 in 1st and 7th positions with 0 as the final digit

I realized that each case would have the same number of permutations, so I would

have to determine only the number of permutations for Case 1.

Step 3. I realized that each sequence can be formed by

• placing the 0 in 10th spot,

• then the 4 in the 1st or 5th spot,

• then the 8 in the remaining spot out of 1st and 5th, and

• then all the remaining digits in the remaining 7 spots.

4.7 Solving Counting Problems

Keep in Mind

In counting problems, use permutations when order is important and use combinations when it is not.

Look for specific features of the problem.

• If there is a repetition of r out of n objects, you should usually divide by r !. • If multiple tasks or stages are linked by the word “and,” apply the

Fundamental Counting Principle (use multiplication). • If multiple tasks or stages are linked by the word “or,” the problem can

usually be broken down into mutually exclusive cases (use addition).

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1114.7 Solving Counting ProblemsNEL

I used the Fundamental Counting Principle:

Number of sequences for Case 1 5 1 ? 2 ? 1 ? 7!, or 10 080

Step 4. The total number of sequences for all three cases is 3 ? (10 080), or 30 240.

Example 2

How many different 4-card hands containing at least 1 heart can be dealt to one person from a standard deck of playing cards?

Solution

Step 1. The order of the cards does not matter, so I knew that the problem involved

combinations. I decided to use indirect reasoning because I realized this would involve

fewer calculations.

Step 2. I chose a strategy of subtracting the number of 4-card hands with no hearts from

the total number of 4-card hands.

Total number of 4-card hands 5 52C4

Number of 4-card hands with no hearts 5 39C4

Number of 4-card hands with at least 1 heart:

52C4 2 39C4 552 ? 51 ? 50 ? 49 ? 48!

4! ? 48!2

39 ? 38 ? 37 ? 36 ? 35!4! ? 35!

5 270 725 2 82 251

5 188 474

There are 188 474 different 4-card hands with at least 1 heart.

Practice

1. Identify whether each situation involves permutations or combinations. Explain how you know.

a) appointing a chair, vice chair, and secretary from a committee of 9 students

b) choosing any 3 of 5 flavours of fruit juice

c) forming a sequence of 1 and 3 symbols using 5 1 symbols and 4 3 symbols

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112 4.7 Solving Counting Problems NEL

2. How many possibilities are there for a five-digit combination using the digits from 0 to 9

a) if each digit can be used many times?

b) if each digit can be used only once?

3. Darryn wants to count the number of sequences he could form using all the spades in a standard deck, with the conditions that

• the king is the final card in the sequence,

• the first card in the sequence is also a face card, and

• the final face card must be 5th or 9th in the sequence.

a) Does this problem involve permutations or combinations? Explain.

b) How many sequences are possible with these conditions?

4. The 16 members of a soccer team are travelling to a match in 3 vehicles, which have room for 7, 5, and 4 passengers, respectively.

a) Does this situation involve permutations or combinations? Explain.

b) Does this situation involve either the Fundamental Counting Principle or the Principle of Inclusion and Exclusion? Explain.

c) In how many ways can the team be seated in the vehicles?

5. Keith is creating a collage using 6 separate prints of an image of Mt. Robson, British Columbia, each print in a different monochrome colour. How many collages using at least 4 of the prints can Keith create?

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6. How many unordered collections of 3 letters and 4 digits are possible if no characters are repeated?

7. In backgammon, counters are placed on points marked by long triangles. You cannot land your counter on a point where your opponent has two or more counters, but you can jump over that point. For example, in the position shown, black can move past the two points held by white with a roll of 1 and 6, but not with a roll of 2 and 5. How many rolls with two standard dice allow black to move past both white points in this position? Count (1, 6) and (6, 1) as different rolls.

NUMERICAL RESPONSE

8. Katie likes to wear odd socks. She can choose two non-matching socks from a drawer with six different, separated pairs of socks in ways.

9. How many arrangements are there of the letters in the word ANAGRAM

a) with no conditions?

arrangements

b) if the arrangements cannot start with an A?

arrangements

WRITTEN RESPONSE

10. How many different 5-card hands containing at most 2 red cards can be dealt to one person from a standard deck of playing cards?

a) Would you use direct or indirect reasoning for this problem? Explain.

b) Determine the number of possible hands with these conditions.

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