Permutations and Combinations Quiz

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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?

  • $ 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?

  • $(n-1)!$ (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?

  • $inom{n}{r} - inom{k}{r}$
  • $inom{n-k}{r-k}$ (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?

<p>$2^n - 1$ (C)</p> Signup and view all the answers

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?

<p>$inom{n-k}{r}$ (D)</p> Signup and view all the answers

What is the result of the sum $inom{n}{0} + inom{n}{1} + inom{n}{2} + insom{n}{3} + insom{n}{n}$?

<p>$2^n$ (C)</p> Signup and view all the answers

How do you calculate the number of ways to form a necklace of n dissimilar beads?

<p>$ rac{(n-1)!}{2}$ (C)</p> Signup and view all the answers

What represents the number of circular permutations of n different things taken r at a time?

<p>$ rac{(n-1)!}{(n-r)!}$ (C)</p> Signup and view all the answers

Flashcards

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

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

The number of unique ways to arrange beads on a necklace, where flipping the necklace is considered the same arrangement.

Combinations with Specific Objects

The number of ways to choose a specific number of objects from a larger set where some objects must always be included.

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Combinations Without Specific Objects

The number of ways to choose a specific number of objects from a larger set where some objects must always be excluded.

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Total Selections

The total number of ways to choose one or more objects from a set. The formula accounts for all possible combinations, from choosing one object to choosing all objects.

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Combinations Formula

The formula that calculates the number of ways to choose r objects from a set of n distinct objects.

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Combinations Sum

The sum of combinations across all possible choices of r from n objects. The sum results in the maximum combinations possible, which is two to the power of n.

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Permutation Formula

The formula that calculates the number of ways to order n objects taken r at a time.

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Study Notes

Permutations and Combinations

  • 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!)

  • Circular Permutations: The number of circular permutations of 'n' different things taken all at once is (n-1)!

  • Necklace Formation: Number of ways to form a necklace with 'n' dissimilar beads is (n-1)! / 2

  • Circular Permutations (r taken at a time): Formula for circular permutations of n different items taken r at a time is not provided.

  • Combinations with Restrictions:

  • 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.

  • 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.

  • 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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