Counting Principles — Practice Quiz

A Probability cheat sheet for Counting Principles — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Factorial

$$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$

Where: \(0! = 1\)

Multiplication Principle

$$\text{Total outcomes} = n_1 \times n_2 \times \cdots \times n_k$$

Where: \(n_i\) = choices at step i

Permutations (order matters)

$$P(n, r) = \frac{n!}{(n-r)!}$$

Combinations (order does not matter)

$$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Permutations with Repetition

$$\frac{n!}{n_1! n_2! \cdots n_k!}$$

Where: \(n_i\) = count of type i

Practice quiz

  1. How many distinct ways can 7 different books be arranged on a shelf?

    • A) $7!$
    • B) $7 \times 6$
    • C) $7^7$
    • D) $7 \times 7$

    Answer: A) $7!$

  2. A car manufacturer offers 5 exterior colors, 3 interior colors, and 2 engine types. How many different car configurations are possible?

    • A) $5+3+2$
    • B) $5 \times 3 \times 2$
    • C) $5^3 \times 2$
    • D) $5! \times 3! \times 2!$

    Answer: B) $5 \times 3 \times 2$

  3. In a competition with 10 participants, how many different ways can the first, second, and third places be awarded?

    • A) $C(10, 3)$
    • B) $P(10, 3)$
    • C) $10^3$
    • D) $10!$

    Answer: B) $P(10, 3)$

  4. A pizza shop offers 12 different toppings. If a customer wants to choose 4 toppings, how many different combinations of toppings are possible?

    • A) $P(12, 4)$
    • B) $12^4$
    • C) $C(12, 4)$
    • D) $12!$

    Answer: C) $C(12, 4)$

  5. How many distinct arrangements can be made from the letters of the word "MATHEMATICS"?

    • A) $\frac{11!}{2!}$
    • B) $\frac{11!}{2!2!}$
    • C) $\frac{11!}{2!2!2!}$
    • D) $11!$

    Answer: C) $\frac{11!}{2!2!2!}$

  6. A license plate consists of 3 distinct letters followed by 3 distinct digits. How many unique license plates are possible? (Assume 26 letters and 10 digits: 0-9)

    • A) $C(26, 3) \times C(10, 3)$
    • B) $P(26, 3) + P(10, 3)$
    • C) $P(26, 3) \times P(10, 3)$
    • D) $26^3 \times 10^3$

    Answer: C) $P(26, 3) \times P(10, 3)$

  7. Which of the following scenarios involves combinations?

    • A) Arranging 5 people in a line.
    • B) Selecting a president, vice-president, and secretary from a group of 10 people.
    • C) Choosing 3 books from a list of 8 to read.
    • D) Forming a 4-digit number using distinct digits.

    Answer: C) Choosing 3 books from a list of 8 to read.

  8. From a standard deck of 52 cards, how many 5-card hands contain exactly 2 Kings and 3 Queens?

    • A) $C(4, 2) + C(4, 3)$
    • B) $C(52, 5)$
    • C) $C(4, 2) \times C(4, 3)$
    • D) $P(4, 2) \times P(4, 3)$

    Answer: C) $C(4, 2) \times C(4, 3)$

  9. How many distinct ways can 4 red balls, 3 blue balls, and 2 green balls be arranged in a row?

    • A) $9!$
    • B) $\frac{9!}{4!3!2!}$
    • C) $C(9, 4) \times C(5, 3) \times C(2, 2)$
    • D) $P(9, 4) \times P(5, 3) \times P(2, 2)$

    Answer: B) $\frac{9!}{4!3!2!}$

  10. A committee of 5 people is to be chosen from a group of 6 men and 4 women. How many ways can the committee be formed if it must include at least 3 women?

    • A) $C(10, 5)$
    • B) $C(4, 3) \times C(6, 2)$
    • C) $C(4, 3) \times C(6, 2) + C(4, 4) \times C(6, 1)$
    • D) $C(4, 3) + C(6, 2)$

    Answer: C) $C(4, 3) \times C(6, 2) + C(4, 4) \times C(6, 1)$

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