Practice Questions

Calculate the value of $8!$.

Compute the value of $\frac{10!}{7!}$.

State the formula for the number of combinations of $n$ different objects taken $r$ at a time, denoted by ${}^n C_r$.

What is the value of $0!$?

Define the term 'permutation'.

Formulate a word problem for which the number of possible outcomes is calculated by the expression ${}^{10}C_4 + {}^6C_4 + {}^4C_4$.

Justify, using a logical argument about selection and rejection, why ${}^nC_r = {}^nC_{n-r}$ without using the factorial formula.

A restaurant offers 5 appetizers, 8 main courses, and 4 desserts. How many different three-course meals, consisting of one appetizer, one main course, and one dessert, can be ordered?

If $^{18}C_r = {^{18}C_{r+2}}$, calculate the value of $r$.

How many 3-letter words (with or without meaning) can be formed from the letters of the word 'LOGARITHM' if repetition of letters is not allowed?

State the formula for the number of permutations of $n$ different objects taken $r$ at a time, denoted by ${}^n P_r$, where repetition is not allowed.

Justify the identity ${}^nP_n = {}^nP_{n-1}$.

If one event can occur in 5 ways and another independent event can occur in 3 ways, in how many ways can both events occur in succession?

Find the value of ${}^8 C_3$.

Calculate the value of $n$ if $(n+1)! = 20 \times (n-1)!$.

Calculate $n$ if $^nC_4 = 35$.

Explain the key difference between a permutation and a combination, providing one example for each.

Find the value of ${}^7 P_3$.

If ${}^n C_4 = {}^n C_6$, recall the property that relates the lower indices and find the value of $n$.

Summarize the relationship between permutations (${}^n P_r$) and combinations (${}^n C_r$). Using this relationship, derive the formula ${}^n C_r = \frac{n!}{r!(n-r)!}$.

Consider the letters of the word 'TOP'. (i) List all possible permutations of the three letters. (ii) List all possible combinations of the three letters taken two at a time.

How many 4-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5 if repetition of digits is not allowed?

A committee of 5 is to be formed from a group of 6 men and 4 women. In how many ways can this be done if the committee must contain at least 2 women?

In a party, every person shakes hands with every other person. If there were a total of 105 handshakes, how many people were present at the party?

From a standard deck of 52 cards, in how many ways can 5 cards be chosen such that there are exactly 2 kings and 2 queens?

Evaluate the statement: "The number of distinct permutations of the letters in the word 'SUCCESS' is greater than the number of 4-letter codes that can be formed from the first 7 letters of the alphabet if no letter is repeated." Justify your conclusion.

Propose a real-world counting problem for which the solution is ${}^5C_2 \times {}^4C_2 \times {}^3C_2$.

A student calculates the number of 5-card hands from a 52-card deck with at least one Ace as ${}^4C_1 \times {}^{51}C_4$. Critique this reasoning, identify the specific flaw, and formulate the correct calculation.

Derive the identity $k \cdot {}^nC_k = n \cdot {}^{n-1}C_{k-1}$ for $1 \le k \le n$ using a combinatorial argument.

Justify why the number of diagonals in a convex $n$-sided polygon is given by the formula ${}^nC_2 - n$.

Prove the identity ${}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$ using a combinatorial argument. Do not use algebraic expansion of factorials.

A committee of 8 members is to be formed from 10 men and 7 women. Formulate a method to find the number of ways this can be done if the committee must contain at least 3 men and at least 3 women. Justify that your method considers all valid cases and avoids overcounting, then calculate the final answer.

Calculate the number of distinct arrangements of the letters in the word 'SUCCESS'.

Explain the concept behind the identity ${}^n C_r = {}^n C_{n-r}$. Then, verify this identity for $n=10$ and $r=3$.

Design a step-by-step method to find the rank of the word 'ZENITH' if all permutations of its letters are listed in dictionary (alphabetical) order. Formulate the steps clearly and use them to calculate the rank.

Describe the formula for finding the number of permutations of $n$ objects, when not all objects are distinct. For example, when there are $p_1$ objects of one kind, $p_2$ objects of a second kind, and so on, up to $p_k$ objects of a $k^{th}$ kind.

Create a set of constraints for arranging the letters of the word 'COMPUTER' such that the total number of distinct arrangements is exactly $6! \times {}^7C_3$.

In how many ways can the letters of the word 'TRIANGLE' be arranged such that the vowels always occupy the odd places?

Create a counting problem involving both permutations and combinations whose final solution is given by the expression ${}^7C_4 \times {}^5C_3 \times \frac{7!}{2!}$. Solve the problem you have created.

A group of 12 students consists of 5 from Science, 4 from Arts, and 3 from Commerce. A committee of 6 is to be formed. Find the number of ways this can be done if: (i) There must be exactly 2 students from each stream. (ii) There must be at least one student from each stream.

There are 15 points in a plane, of which 4 are collinear. How many distinct straight lines can be formed by joining these points?

In how many ways can the letters of the word 'DIRECTOR' be arranged so that the three vowels (I, E, O) are never together?

Evaluate which quantity is greater and justify your answer with calculations: A) The number of ways to arrange 5 boys and 4 girls in a row so that no two girls are together. B) The number of ways to select a team of 4 from the 9 individuals and then elect a captain and vice-captain from the selected team.

Critique the following reasoning: "To find the number of ways to arrange 4 boys and 3 girls in a row so that all 3 girls sit together, we choose 3 spots for the girls in ${}^7C_3$ ways and arrange them in $3!$ ways. Then we arrange the 4 boys in the remaining 4 spots in $4!$ ways. The answer is ${}^7C_3 \times 3! \times 4!$."