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Questions and Answers
What is the formula for the total number of permutations of p1 things alike of one kind, p2 alike of a second kind, and so on, up to pk things alike?
What is the formula for the total number of permutations of p1 things alike of one kind, p2 alike of a second kind, and so on, up to pk things alike?
- $rac{(p_1 + p_2 +... + p_k)!}{p_1! + p_2! + ... + p_k!}$
- $rac{(p_1 + p_2 +... + p_k)!}{p_1!p_2!...p_k!}$ (correct)
- $(p_1 + p_2 + ... + p_k)!/(p_1p_2...p_k)$
- $(p_1 + p_2 + ... + p_k)!$
What is the number of circular permutations of n different things taken all at a time?
What is the number of circular permutations of n different things taken all at a time?
- $(n-1)!$ (correct)
- $n!$
- $rac{n!}{2}$
- $(n+1)!$
How is the number of combinations of n distinct objects taken r at a time calculated when k specific objects always occur?
How is the number of combinations of n distinct objects taken r at a time calculated when k specific objects always occur?
- $inom{n}{r} - inom{k}{r}$
- $inom{n-k}{r-k}$ (correct)
- $inom{n}{r-k}$
- $inom{n-k}{r}$
What is the total number of selections of one or more objects from n different objects?
What is the total number of selections of one or more objects from n different objects?
What is the correct expression for the number of combinations of n distinct objects taken r at a time when k specific objects never occur?
What is the correct expression for the number of combinations of n distinct objects taken r at a time when k specific objects never occur?
What is the result of the sum $inom{n}{0} + inom{n}{1} + inom{n}{2} + insom{n}{3} + insom{n}{n}$?
What is the result of the sum $inom{n}{0} + inom{n}{1} + inom{n}{2} + insom{n}{3} + insom{n}{n}$?
How do you calculate the number of ways to form a necklace of n dissimilar beads?
How do you calculate the number of ways to form a necklace of n dissimilar beads?
What represents the number of circular permutations of n different things taken r at a time?
What represents the number of circular permutations of n different things taken r at a time?
Flashcards
Permutations of alike objects
Permutations of alike objects
The number of ways to arrange a set of objects where some objects are identical. The formula accounts for the fact that swapping identical objects doesn't create a new arrangement.
Circular Permutations
Circular Permutations
Arrangements of objects in a circle where rotations are considered the same arrangement. The formula accounts for the fact that rotating the circle doesn't create a new arrangement.
Necklace Permutations
Necklace Permutations
The number of unique ways to arrange beads on a necklace, where flipping the necklace is considered the same arrangement.
Combinations with Specific Objects
Combinations with Specific Objects
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Combinations Without Specific Objects
Combinations Without Specific Objects
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Total Selections
Total Selections
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Combinations Formula
Combinations Formula
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Combinations Sum
Combinations Sum
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Permutation Formula
Permutation Formula
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Study Notes
Permutations and Combinations
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Identical Objects: If 'p' things are alike of one kind, 'p₂' alike of a second kind, and so on, the number of permutations is (p₁+p₂+p₃...+pₖ)! / (p₁! p₂! p₃!...pₖ!)
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Circular Permutations: The number of circular permutations of 'n' different things taken all at once is (n-1)!
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Necklace Formation: Number of ways to form a necklace with 'n' dissimilar beads is (n-1)! / 2
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Circular Permutations (r taken at a time): Formula for circular permutations of n different items taken r at a time is not provided.
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Combinations with Restrictions:
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If k particular objects always occur in a combination of n distinct objects taken r at a time (0 ≤ k ≤ r), the number of combinations is n-kCr-k.
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If k objects never occur in a combination of n distinct objects taken r at a time (1 ≤ k ≤ r), the number of combinations is n-kCr.
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Total Selections: The total number of selections of one or more objects from n different objects is 2ⁿ - 1 = nC₁ + nC₂ + nC₃ +...+ nCn
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