matura.mkSecondary school › Counting and probability
The binomial theorem
Expanding binomials, a given term, the constant term and binomial identities. 7 problems with step-by-step solutions.
Solve each problem in your notebook first, then open the solution.
Problem 1 Expanding binomials
Expand.
a) \((x + 2y)^4\) b) \((2a - b)^5\) c) \(\left(x - \frac{1}{x}\right)^4\) d) \((1 + \sqrt3)^4\)
Show solution
Coefficients for powers 4 and 5: 1, 4, 6, 4, 1 and 1, 5, 10, 10, 5, 1.
a) \(x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4\)
b) \(32a^5 - 80a^4b + 80a^3b^2 - 40a^2b^3 + 10ab^4 - b^5\)
c) \(x^4 - 4x^2 + 6 - \frac{4}{x^2} + \frac{1}{x^4}\)
d) \(1 + 4\sqrt3 + 18 + 12\sqrt3 + 9 = 28 + 16\sqrt3\)
Problem 2 Number of terms and sums
a) How many terms are there in the expansion of \((a + b)^{15}\)?
b) Find the sum of the coefficients of \((2x - 3y^2)^5\).
c) Calculate \(\binom60 + 3\binom61 + 3^2\binom62 + \dots + 3^6\binom66\).
Show solution
a) 16.
b) Put \(x = y = 1\): \((2 - 3)^5 = -1\).
c) \((1 + 3)^6 = 4096\).
Problem 3 A given term
The \((r + 1)\)-th term of \((x + y)^n\) is \(\binom{n}{r}x^{n - r}y^r\).
a) Find the 4th term of \((a + b)^{10}\). b) Find the 5th term of \((x + 2)^8\).
c) Find the coefficient of the 3rd term of \((3x - y)^6\). d) Find the middle term of \((x^2 + 2y)^6\).
Show solution
a) \(\binom{10}{3}a^7b^3 = 120a^7b^3\).
b) \(\binom84x^4\cdot 2^4 = 1120x^4\).
c) \(\binom62 3^4(-1)^2 = 15\cdot 81 = 1215\).
d) 7 terms, the middle is the 4th: \(\binom63(x^2)^3(2y)^3 = 160x^6y^3\).
Problem 4 Constant term
a) Find the constant term of \(\left(x^2 + \frac{3}{x}\right)^6\).
b) Find the constant term of \(\left(2x - \frac{1}{x^2}\right)^9\).
c) Find the constant term of \((5x + 2)^4\).
Show solution
a) General term \(\binom6r x^{2(6 - r)}3^rx^{-r} = \binom6r 3^r x^{12 - 3r}\). \(r = 4\): \(15\cdot 81 = 1215\).
b) \(\binom9r(2x)^{9 - r}(-1)^rx^{-2r}\): power \(9 - 3r = 0\), \(r = 3\): \(84\cdot 2^6\cdot(-1) = -5376\).
c) \(2^4 = 16\).
Problem 5 Coefficient of a power
a) Find the coefficient of \(x^3y^2\) in \((x - 2y)^5\).
b) \(160x^3y^3\) is a term of \((ax + 2y)^6\). Find \(a\).
c) Find the terms of the expansion of \((\sqrt2 + \sqrt[3]{3})^6\) that are integers.
Show solution
a) \(\binom52(-2)^2 = 40\).
b) \(\binom63a^3\cdot 8 = 160a^3 = 160\), so \(a = 1\).
c) Term \(\binom6r 2^{(6 - r)/2}3^{r/3}\) is an integer when \(6 - r\) is even and \(r\) is a multiple of 3: \(r = 0\) or \(r = 6\). Terms: \(2^3 = 8\) and \(3^2 = 9\).
Problem 6 Binomial identities
a) Show that \(\binom{n}{0} + \binom n1 + \dots + \binom nn = 2^n\).
b) Calculate \(\binom{10}{0} - \binom{10}{1} + \binom{10}{2} - \dots + \binom{10}{10}\).
c) List the coefficients of the 8th power.
Show solution
a) Put \(x = y = 1\) in \((x + y)^n\).
b) \((1 - 1)^{10} = 0\).
c) 1, 8, 28, 56, 70, 56, 28, 8, 1.
Problem 7 Mixed counting test
a) Simplify \(\dfrac{(n - 9)!\,(n + 10)!}{(n - 10)!\,(n + 9)!}\).
b) How many 4-digit numbers with different digits can be formed from \(\{0, 1, 2, 3, 4\}\)?
c) In how many ways can 2 couples and 3 single people sit at a round table if each couple sits together?
d) How many different words can be formed from all the letters of BALLOON?
Show solution
a) \((n - 9)(n + 10)\).
b) \(4\cdot 4\cdot 3\cdot 2 = 96\).
c) 2 blocks + 3 people = 5 objects: \(4!\cdot 2\cdot 2 = 96\).
d) \(\frac{7!}{2!\,2!} = 1260\).