Standard +0.3 This is a straightforward binomial coefficient problem requiring students to set up an equation using $inom{n}{4} = rac{1}{2}[inom{n}{2} + inom{n}{3}]$, expand the binomial coefficients into factorial form, and solve a quadratic equation. While it involves algebraic manipulation across multiple steps, it's a standard textbook-style question with no novel insight required, making it slightly easier than average.
In the binomial expansion of \((1 + x)^n\), where \(n \geq 4\), the coefficient of \(x^4\) is \(\frac{1}{2}\) times the sum of the coefficients of \(x^2\) and \(x^3\)
Find the value of \(n\).
[5 marks]
Question 10:
10 | The 3 appropriate C seen (ignore
n r
any extras) (PI by 55, 165 and 330
OE) | AO3.1a | M1 | C C C
n 2 n 3 n 4
3 𝑛! 𝑛!
( + )
2 (𝑛−2)!2! (𝑛−3)!3!
𝑛!
=
(𝑛−4)!4!
3 𝑛(𝑛−1) 𝑛(𝑛−1)(𝑛−2)
( + )
2 2 6
𝑛(𝑛−1)(𝑛−2)(𝑛−3)
=
24
18+6𝑛−12 = 𝑛2−5𝑛+6
0 = 𝑛2−11𝑛 = 𝑛(𝑛−11)
𝑛 = 11
Forms a correct equation, accept
3
( C + C ) = C
2 n 2 n 3 n 4
𝑛
Allow ( ) notation
𝑟
(condone x terms in equation) | AO1.1a | M1
Obtains completely correct
equation in terms of factorials.
Reaching second line of typical
solution scores M1 M1 A1 | AO1.1b | A1
Reduces to a quadratic or solves
the quartic (may involve calculator
functions) | AO1.1a | M1
Chooses the correct solution.
(The correct value of n scores 5/5
may be found by trial and error) | AO3.2a | A1
Total | 5
Q | Marking Instructions (I) | AO | Marks | Typical Solution (using r)
In the binomial expansion of $(1 + x)^n$, where $n \geq 4$, the coefficient of $x^4$ is $\frac{1}{2}$ times the sum of the coefficients of $x^2$ and $x^3$
Find the value of $n$.
[5 marks]
\hfill \mbox{\textit{AQA AS Paper 2 2018 Q10 [5]}}