Counting Principles — Hard Practice Quiz

A Probability cheat sheet for Counting Principles — every key formula with its symbols defined — plus a hard-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. A committee of $5$ members is to be formed from a group of $8$ men and $6$ women. The committee must include at least $3$ women. Once the committee is formed, a president and a vice-president must be selected from these $5$ members. How many distinct ways can this entire process be completed?

    • $13720$
    • $6860$
    • $3430$
    • $27440$

    Answer: $13720$

  2. How many distinct arrangements of the letters in the word "MATHEMATICS" are possible if the arrangement must begin with 'M' and end with 'S'?

    • $90720$
    • $181440$
    • $362880$
    • $45360$

    Answer: $90720$

  3. A security code consists of $7$ characters. The first $3$ characters must be distinct digits ($0-9$). The next $2$ characters must be distinct uppercase letters ($A-Z$). The last $2$ characters can be any digit or uppercase letter, but they cannot repeat any of the first $5$ characters used. How many unique security codes are possible?

    • $435240000$
    • $388800000$
    • $504000000$
    • $405600000$

    Answer: $435240000$

  4. Given that $P(n, r) = 1680$ and $C(n, r) = 70$, determine the values of $n$ and $r$.

    • $n=8, r=4$
    • $n=7, r=4$
    • $n=8, r=3$
    • $n=7, r=3$

    Answer: $n=8, r=4$

  5. A team of $6$ is to be selected from $9$ men and $7$ women. How many different teams can be formed if the team must include at least $2$ men and at least $3$ women?

    • $4200$
    • $3675$
    • $4550$
    • $5040$

    Answer: $4200$

  6. How many distinct permutations of the letters in the word "ENGINEERING" are there such that all three 'E's are not together?

    • $262080$
    • $277200$
    • $15120$
    • $250000$

    Answer: $262080$

  7. A student must answer $10$ questions out of $13$ on an exam. If the first $4$ questions are compulsory, and the student must answer exactly $3$ of the last $5$ questions, how many ways can the student choose the questions?

    • $40$
    • $60$
    • $80$
    • $120$

    Answer: $40$

  8. If $C(n, 3) = 56$, what is the value of $P(n, 4)$?

    • $1680$
    • $840$
    • $3360$
    • $2520$

    Answer: $1680$

  9. A company has $15$ employees: $6$ senior staff and $9$ junior staff. A project team of $5$ members is to be formed. How many ways can the team be formed if it must include at least $2$ senior staff and at most $3$ junior staff?

    • $2121$
    • $1980$
    • $2340$
    • $2050$

    Answer: $2121$

  10. A bookshelf has $4$ distinct novels, $3$ distinct history books, and $2$ distinct science books. In how many ways can these $9$ books be arranged on a shelf if all books of the same subject must be kept together?

    • $1728$
    • $864$
    • $288$
    • $144$

    Answer: $1728$

Select a subject

Select a subject from the left panel to begin exploring formulas.