📝 Chapter 3: Combinations

Compute \( \binom{12}{4} \).

Use the symmetry property to evaluate \( \binom{15}{13} \).

Find the constant term of \( (x+1)^6 \) when \( x=1 \).

Solve \( \binom{n}{2} = 28 \).

In the expansion of \( (2x - 3)^4 \), what is the coefficient of \( x^2 \)?

From a class of 25 students (12 boys, 13 girls), how many committees of 5 with exactly 3 girls are possible?

A poker hand of 5 cards is dealt from a 52-card deck. How many possible hands are there?

A test has 20 multiple-choice questions and a student must answer exactly 15. In how many ways can the student choose? If question 1 must be answered, how does the count change?

A 5-card poker hand is dealt from a standard deck. Determine the probability of getting exactly 3 hearts. Show all work as a single fraction or decimal.

In how many ways can a school select 4 representatives from 12 students if at least 2 must be from a specific group of 5? Hint: Use case analysis on the number selected from the group.

Use the binomial theorem to find the term containing \( x^4 \) in the expansion of \( \left(2x + \dfrac{1}{x}\right)^6 \).

Explain in your own words the difference between a permutation and a combination. Use a small example with 3 objects to show why \( P(3,2) \ne C(3,2) \).

A student says, "Pascal's triangle is useful but Pascal's identity \( \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \) is just a coincidence." Disprove this by giving a clear combinatorial argument (consider counting the ways to choose \( r+1 \) items from \( n+1 \) items based on whether a particular item is selected).

Describe step-by-step how to determine the coefficient of a specific term in the expansion of \( (a+b)^n \) using the binomial theorem. Apply your steps to find the coefficient of \( x^3 \) in \( (x+2)^7 \).

Compare and contrast \( \binom{n}{r} \) and \( P(n,r) \). Write the formulas, and explain why \( P(n,r) = r! \cdot \binom{n}{r} \).

In Lotto 6/49, players choose 6 distinct numbers from 1 to 49. a) How many possible tickets are there? b) What is the probability of winning the jackpot with one ticket? c) If a ticket costs $3 and the jackpot is $5 million, is this a fair bet (compare expected value to ticket price)?

A box of 20 light bulbs has 4 defective. A sample of 5 is drawn. Find the probability that exactly 1 is defective. Use combinations.

A pizza shop offers 12 toppings. How many different pizzas with 3, 4, or 5 toppings are possible (one of each combination)?

A school's science fair has 25 projects. The judges award 1st, 2nd, and 3rd prizes (gold, silver, bronze) and select 5 honourable mentions. In how many ways can the prizes be assigned (no project can win more than one)?