Permutations and Combinations
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Questions and Answers

If event X can be decomposed into three stages A, B, and C, where stage A occurs in m ways, stage B occurs in n ways, and stage C occurs in p ways, then the total number of ways event X can occur is:

  • m + n + p
  • m × (n + p)
  • m × n × p (correct)
  • n × (m + p)

In a quiz, a student has to answer 5 questions from a set of 10 questions. In how many ways can the student select the questions?

  • 10 × 5
  • 10 × 9 × 8 × 7
  • 10 × 9 × 8 × 7 × 6
  • 10 × 9 × 8 × 7 × 6 / 5! (correct)

A committee of 3 people is to be formed from a group of 12 people. In how many ways can the committee be formed?

  • 12 × 3
  • 12 × 11 × 10
  • 12 × 11 × 10 × 3
  • 12 × 11 × 10 / 3! (correct)

What is the number of ways to select exactly one course from 7 morning courses and 5 evening courses?

<p>7 + 5 (B)</p> Signup and view all the answers

A password consists of 5 characters, where each character can be one of 26 letters or 10 digits. In how many ways can the password be formed?

<p>36^5 (D)</p> Signup and view all the answers

A bookshelf has 5 shelves, and each shelf can hold 8 books. In how many ways can the books be arranged on the shelves?

<p>8^5 (D)</p> Signup and view all the answers

If a set of objects has 'n' items, and 'p' is a prime number, what is the value of Ep(n!)?

<p>(n!)/p + (n!/p^2) + (n!/p^3) +... (C)</p> Signup and view all the answers

What is the fundamental principle of counting used to find the number of ways to select exactly one course from 7 morning courses and 5 evening courses?

<p>Addition Principle (D)</p> Signup and view all the answers

A team of 3 people is to be formed from a group of 12 people, where the order of selection matters. In how many ways can the team be formed?

<p>12 × 11 × 10 (C)</p> Signup and view all the answers

If a set of objects has 'n' items, and 'p' is a prime number, what is the number of ways to arrange 'n' items in a circular pattern, if 'p' of them are identical?

<p>n!/p (D)</p> Signup and view all the answers

What is the term for the arrangement of objects in a specific order, where the order of the objects matters?

<p>Permutation (D)</p> Signup and view all the answers

What is the principle used to find the total number of ways to select one course from either morning or evening courses, if the morning and evening courses are mutually exclusive?

<p>Addition Principle (A)</p> Signup and view all the answers

What is the value of 0!?

<p>1 (A)</p> Signup and view all the answers

What is the value of n! when n is a negative integer?

<p>not defined (D)</p> Signup and view all the answers

What is the exponent of the prime p in n!?

<p>the largest integer less than or equal to n/p (A)</p> Signup and view all the answers

What is the value of 5!??

<p>5 × 4! (A)</p> Signup and view all the answers

What is the formula for n!?

<p>n × (n-1) × (n-2) ×... × 3 × 2 × 1 (D)</p> Signup and view all the answers

What is the value of E_p(n) in terms of n and p?

<p>the largest integer less than or equal to n/p + E_p(n/p) (B)</p> Signup and view all the answers

How many ways can 3 different things be arranged from a set of 3 things?

<p>6 (A)</p> Signup and view all the answers

What is the formula for the number of permutations of n objects taken r at a time?

<p>n(n-1)(n-2)...(n-r+1) (B)</p> Signup and view all the answers

If nP4 : nP5 = 1 : 2, then what is the value of n?

<p>6 (D)</p> Signup and view all the answers

What is the number of ways to arrange n different objects taken all at a time?

<p>n! (B)</p> Signup and view all the answers

What is the term for the arrangement of objects in a specific order, where the order of the objects matters?

<p>Permutation (B)</p> Signup and view all the answers

If m objects are alike of one kind and the remaining (n-m) objects are alike of another kind, what is the total number of permutations of these n objects?

<p>n! / (m! × (n-m)!) (C)</p> Signup and view all the answers

What is the number of ways to arrange 5 objects, where 2 objects are always together?

<p>5! / 2! (A)</p> Signup and view all the answers

In how many ways can 5 males and 3 females be arranged in a row such that no two females are together?

<p>5! × 3! × 6P3 (D)</p> Signup and view all the answers

What is the formula for the number of permutations of n objects, where p1 are alike of one kind, p2 are alike of another kind, and so on until pr are alike of the rth kind?

<p>n! / (p1! × p2! ×... × pr!) (A)</p> Signup and view all the answers

What is the number of ways to arrange 5 objects in a circular pattern, where 2 objects are identical?

<p>(5-1)! / 2! (A)</p> Signup and view all the answers

What does the notation â‚‹â‚™C â‚š or C(â‚™, â‚š) represent?

<p>The number of combinations of n objects taken r at a time. (D)</p> Signup and view all the answers

What is an example of a combination?

<p>Selecting 2 out of 3 persons A, B, and C. (B)</p> Signup and view all the answers

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