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PRACTICE PROBLEMS: Solve the following problems: 1. Using the digits 4, 5, 6, and 7, how many twodigit
numbers can be formed a. without repetition? b. with repetition? 2. Using the digits 4, 5, 6, 7, 8, and 9, how many
fivedigit numbers can be formed a. without repetition? b. with repetition? 3. Using the digits of problem 2, how many fourdigit odd
numbers can be formed without repetition? ANSWERS:
SUMMARY The following are the major topics covered in this
chapter: 1. Definitions: A combination is defined as a possible selection
of a certain number of objects taken from a group without regard to order. A permutation is defined as a possible selection
of a certain number of objects taken from a group with regard to order. The product of the integers n through
1 is defined as n factorial, and the symbol n! is used to
denote this. 2. Factorial: Theorem. If n and r are positive
integers, with n greater than r, then
3. Combination formula:
for the number of combinations of n objects taken r at a time. 4. Principle of Choice: If a selection can be made in n_{1} ways;
and after this selection is made, a second selection can be made in n_{2} ways; and
after this selection is made, a third selection can be made in n_{3}
ways; and so forth for r selections,
then the sequence of r selections can be made together in _{
}ways. 5. Permutation formula: ^{} for the number of permutations of n objects
taken r at a time. 6. Arrangements: The number of arrangements of n items,
where there are k groups of like items of size r_{1}, r_{2}, r_{k},
respectively, is given by
7. Repetition: Combinations and permutation problems, with
or without repetition, may be solved for using position notation instead of
formulas. ADDITIONAL PRACTICE PROBLEMS 1.
Find the value of
2.
Simplify
. 3.
Find the value of ^{
} 4.
On each trip, a salesman visits 4 of the 12 cities in his territory. In how
many different ways can he schedule his route? 5. From six men and five women, find the number of groups
of four that can be formed consisting of two men and two women. 6. Find the value of
7. In how many ways can the 18 members of a boy scout
troop elect a president, a vicepresident, and a secretary, assuming that no
member can hold more than one office? 8. How many different ways can 4 red, 3 blue, 4 yellow,
and 2 green bulbs be arranged on a string of Christmas
tree lights with 13 sockets? 9. How many car tags can be made if the first three
positions are letters and the last three positions are numbers (Hint:
Twentysix letters and ten distinct singledigit numbers are possible) a. with repetition? b. without repetition? ANSWERS
TO ADDITIONAL PRACTICE PROBLEMS

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