7.04 Binomial theorem
The 4th row of Pascal’s triangle consists of the numbers 1, 4, 6, 4, 1.
Write down the numbers in the 5th row of Pascal’s triangle.
Find the sum of the entries in the 5th row of Pascal’s triangle.
Rewrite each element of the 5th row of Pascal’s triangle as an {}^{n}C_{r}coefficient.
Complete the following entries in a particular row of Pascal’s triangle: 1,8,⬚,56,70,⬚, 28,⬚,1
Which row of Pascal’s triangle would give us the coefficients of terms in the expansion of \left(x + y\right)^{7}?
State the number of terms in the following expansions:
\left( 4 x + y\right)^{6}
\left(m + y\right)^{8}
Consider the expansion of \left(y + k\right)^{4}.
How many terms will there be in the expansion?
Write the coefficient of the 2nd term.
Using the relevant row of Pascal’s triangle, determine the coefficient of each term in the following expansion:
\left(x + m\right)^{4} = ⬚ x^{4} m^{0} + ⬚ x^{3} m^{1} + ⬚ x^{2} m^{2} + ⬚ x^{1} m^{3} + ⬚ x^{0} m^{4}
Evaluate the following:
\binom{6}{5}
\binom{14}{1}
\binom{10}{9}
\binom{8}{0}
Find the specified term for the following expansions:
Fifth term of \left(b + 2\right)^{7}.
Fifth term of \left(c + d\right)^{9}.
Fifth term of (3u-v)^{6}.
Tenth term of \left(u + \dfrac{1}{2}\right)^{12}.
Seventh term of \left(\dfrac{3 y}{2} - \dfrac{2}{3 y}\right)^{10}.
Eleventh term of \left( 2 x + y^{2}\right)^{13}.
Sixteenth term of \left(x - y^{5}\right)^{19}.
Middle term of \left( 4 x^{2} + 3 y^{3}\right)^{8}.
Middle term of \left( - 3 x^{ - 3 } + 2 y^{ - 2 }\right)^{4}.
Constant term of \left( 2 y - \dfrac{1}{y^{3}}\right)^{8}.
The first three terms of \left(x + y\right)^{10} are x^{10}, 10 x^{9} y and 45 x^{8} y^{2}. Using the symmetry of the coefficients, write the last three terms of the expansion.
In the expansion of \left(m + k\right)^{n}, the coefficient of the 9th term is 45.
Find the value of n.
Find the 9th term.
A particular term in the expansion of \left( 3 a^{2} + \dfrac{p}{b}\right)^{4} is \dfrac{96 a^{2}}{b^{3}}, for some constant p. Find the value of p.
Consider the expansion of \left(a - 4\right)^{5}. Will the 2nd term be positive or negative?
By considering \left(x + y\right)^{4} = \left(x + y\right) \left(x + y\right) \left(x + y\right) \left(x + y\right), in how many ways can three factors of x and one factor of y be chosen from the expansion to form the term which contains x^{3} y^{1}? Write your answer in the form {}^{n}C_{r}.
Complete the following expansion:
\left( 4 p + 3 q\right)^{3} = {}^{3}C_{0} \left( 4 p\right)^{⬚} \left( 3 q\right)^{0} + {}^{3}C_{1} \left( 4 p\right)^{⬚} \left( 3 q\right)^{1} + {}^{3}C_{2} \left( 4 p\right)^{⬚} \left( 3 q\right)^{2} + {}^{3}C_{3} \left( 4 p\right)^{⬚} \left( 3 q\right)^{3}
Consider the binomial \left( 4 x + 3 y\right)^{4}.
State the first term in the expansion.
State the last term in the expansion.
Consider the expansion of \left(a + b\right)^{5}.
State the coefficient of the 3rd term in the form \binom{n}{r}.
Evaluate the coefficient of the 3rd term.
By considering the symmetry of \binom{n}{r}, which other term has the same coefficient as the 3rd term?
Which two terms in the expansion of \left(u + v\right)^{11} have a coefficient of \binom{11}{9}?
Find the coefficient of x^{17} in the expansion of \left( 3 x^{2} + 2 x\right)^{11}.
Use the binomial theorem to expand the following expressions:
\left(1 + x\right)^{4}
\left(2 + b\right)^{5}
\left(p + q\right)^{5}
\left(c - d\right)^{8}
\left(u + 4 v\right)^{6}
\left(y + 3\right)^{4}
( k a - b)^{6}
\left(y - 5\right)^{3}
\left(y - \dfrac{1}{3}\right)^{3}
\left(2 + \dfrac{y}{2}\right)^{3}
\left( 3 y + 2 r\right)^{3}
\left(a - \dfrac{1}{a}\right)^{3}
\left(u^{2} + 3 v^{2}\right)^{3}
\left(1 + \dfrac{4}{y}\right)^{3}
Find:
The term in \left(3 + \sqrt{x}\right)^{10} that contains x^{4}.
The term in \left(c + d\right)^{10} that contains c^{3}.
The term in \left(u + v\right)^{11} that contains v^{8}.
The term in \left(u + v\right)^{15} that contains u^{5}.
If the eighth and tenth terms in the expansion of \left(x + y\right)^{n} have the same coefficients, find the value of n.
Find the coefficient of a^{3} b^{6} in the expansion of \left( 2 a - \dfrac{b}{2}\right)^{9}.
Consider the expansion of \left(1 + x\right)^{9}. State the coefficient of the 4th term in the form \binom{n}{r}.
Find the value of n such that the expansion of \left(a + b\right)^{n} contains the term 84 a^{6} b^{3}.
Complete the following expansion by determining the missing power and binomial coefficient of the \left(k + 1\right)th term:
\left(3 - x\right)^{8} = {}^{8}C_{0} \times 3^{8} \left( - x \right)^{0} + {}^{8}C_{1} \times 3^{7} \left( - x \right)^{1}+\ldots+⬚ \times 3^{8 - k} \left( - x \right)^{⬚}+\ldots+{}^{8}C_{8} \times 3^{0} \left( - x \right)^{8}
Consider the binomial series:
\left(1 + x\right)^{n} = 1 + n x + \dfrac{n \left(n - 1\right)}{2!} x^{2} + \dfrac{n \left(n - 1\right) \left(n - 2\right)}{3!} x^{3} + \ldots
Use the first four terms to approximate \left(1.03\right)^{ - 2 } to the nearest thousandth.
Use the first four terms to approximate \left(1.05\right)^{\frac{3}{4}} to the nearest thousandth.
Use the first five terms to approximate \dfrac{1}{\left(1.06\right)^{2}} to the nearest thousandth.