1. Counting Examples
Mathematics 4
February 14, 2012
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2. Fundamental Counting Principle
If one event can be done independently n different ways and a second
event can be done independently m different ways, the total number of
ways all things may be done in the stated order is n · m.
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3. Example 1
How many three-digit numbers can be formed from the digits 1, 2, 3, 4,
and 5 if repetitions are allowed? If repetitions are not allowed?
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4. Example 2
How many numbers can be formed from the digits 1, 2, 3, 4, and 5 if
repetitions are not allowed? (1 to 54321)
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5. Example 3
If eight basketball teams are in a tournament, find the number of different
ways that first, second, and third place can be decided, assuming ties are
not allowed.
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6. Example 4
A girl has four skirts, six blouses, and three sweaters. How many different
skirt-blouse-sweater combinations can she wear?
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7. Example 5
Automobile license plates start with three letters of the alphabet, followed
by three digits. Find how many different license plates are possible if
1 the first digit following the last letter cannot be 0
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8. Example 5
Automobile license plates start with three letters of the alphabet, followed
by three digits. Find how many different license plates are possible if
1 the first digit following the last letter cannot be 0
2 the first letter cannot be I or O and the first digit cannot be 0
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9. Example 6
Two diced are tossed, one after the other. In how many different ways can
they fall?
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10. Example 7
Two diced are tossed, one after the other. How many different ways can
the sum of the dots equal
1 3
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11. Example 7
Two diced are tossed, one after the other. How many different ways can
the sum of the dots equal
1 3
2 5
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12. Example 7
Two diced are tossed, one after the other. How many different ways can
the sum of the dots equal
1 3
2 5
3 7
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13. Example 7
Two diced are tossed, one after the other. How many different ways can
the sum of the dots equal
1 3
2 5
3 7
4 9
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14. Example 7
Two diced are tossed, one after the other. How many different ways can
the sum of the dots equal
1 3
2 5
3 7
4 9
5 11
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15. Example 8
A row of six seats in a classroom is to be filled by selecting individuals
from a group of six boys and four girls. If boys and girls are to be
alternated, find the number of different seating arrangements.
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16. Example 9
In how many ways can a test consisting of ten true-or-false questions can
be completed?
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17. Example 10
A test consists of six multiple-choice questions, and there are five choices
for each question. In how many different ways can the test be completed?
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18. Example 11
Three high-school couples have purchased movie tickets. Partners are to
be seated next to each other, and the six seats are in a row. In how many
ways can the six people be seated?
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19. Example 12
In how many ways can a standard deck of cards be shuffled so that the
four aces appear on the top of the deck?
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