📝 Chapter 3: Combinations

Assessment AS Learning — Practice Quiz
🔄 Not Graded — Unlimited Retakes
Purpose: Self-check your skills with combinations \( C(n,r) \), Pascal's triangle, the binomial theorem, and combinations in probability.
Score: 0 / 12
Topic 3.1 — Combinations \( C(n,r) \)
Question 1
Evaluate \( \binom{8}{3} \).


Solution:
\( \binom{8}{3} = \dfrac{8!}{3!\,5!} = \dfrac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56 \).
Question 2
A committee of 4 is to be selected from 10 people. How many possible committees?


Solution:
\( \binom{10}{4} = \dfrac{10!}{4!\,6!} = \dfrac{10\cdot 9 \cdot 8 \cdot 7}{4!} = \dfrac{5040}{24} = 210 \).
Question 3
From a deck of 52 cards, how many 5-card hands are possible?


Solution:
\( \binom{52}{5} = \dfrac{52!}{5!\,47!} = 2\,598\,960 \).
Topic 3.2 — Combination Identities
Question 4
Use the identity \( \binom{n}{r} = \binom{n}{n-r} \) to evaluate \( \binom{20}{18} \) more easily.


Solution:
\( \binom{20}{18} = \binom{20}{2} = \dfrac{20\cdot 19}{2} = 190 \).
Question 5
Verify Pascal's identity: \( \binom{6}{2} + \binom{6}{3} \) equals which of the following?
Solution:
Pascal: \( \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \). So \( \binom{6}{2}+\binom{6}{3} = \binom{7}{3} = 35 \). Check: \( 15 + 20 = 35 \). ✓
Topic 3.3 — Pascal's Triangle
Question 6
What is the sum of the entries in row 5 of Pascal's triangle (where row 0 has just \( \binom{0}{0} \))?


Solution:
The sum of row \( n \) is \( 2^n \). Row 5 sum = \( 2^5 = 32 \). (Row 5 = 1, 5, 10, 10, 5, 1.)
Topic 3.4 — Binomial Theorem
Question 7
In the expansion of \( (x + 2)^5 \), what is the coefficient of \( x^3 \)?


Solution:
General term: \( \binom{5}{k} x^{5-k} 2^k \). For \( x^3 \): \( 5-k=3 \Rightarrow k=2 \). Coefficient: \( \binom{5}{2} \cdot 2^2 = 10 \cdot 4 = 40 \).
Question 8
Find the constant term in the expansion of \( (a+b)^4 \) when \( a = b = 1 \).
Solution:
\( (1+1)^4 = 2^4 = 16 \). Equivalently, sum of row 4 of Pascal's triangle: 1+4+6+4+1 = 16.
Topic 3.5 — Combinations & Probability
Question 9
A 5-card hand is drawn from a 52-card deck. What is the probability that all 5 cards are Hearts? (Express as \( \dfrac{a}{b} \) reduced; enter approximate decimal to 5 places.)


Solution:
\( P = \dfrac{\binom{13}{5}}{\binom{52}{5}} = \dfrac{1287}{2\,598\,960} \approx 0.000495 \).
Question 10
A bag has 5 red, 4 blue, 3 green marbles. Three are drawn at random. Find \( P(\text{exactly 2 red}) \). (Decimal to 4 places)


Solution:
\( P = \dfrac{\binom{5}{2}\binom{7}{1}}{\binom{12}{3}} = \dfrac{10 \cdot 7}{220} = \dfrac{70}{220} \approx 0.3182 \). (Accept 0.3182–0.3190 due to rounding.)
Mixed
Question 11
In how many ways can a teacher select 3 from 8 boys AND 2 from 6 girls to form a 5-person team?


Solution:
\( \binom{8}{3} \cdot \binom{6}{2} = 56 \cdot 15 = 840 \).
Question 12
Solve for \( n \): \( \binom{n}{2} = 21 \).


Solution:
\( \dfrac{n(n-1)}{2} = 21 \Rightarrow n(n-1) = 42 \Rightarrow n^2 - n - 42 = 0 \Rightarrow (n-7)(n+6)=0 \Rightarrow n = 7 \).