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Counting & Binomials
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What techniques do we have for counting mathematical objects?

Key questions

  1. 1

    How can we use symmetry to help us to count?

  2. 2

    How can we count permutations and combinations?

  3. 3

    How can we expand \((x+y)^n\) efficiently?

  4. 4

    How can we use logical reasoning to solve problems?

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Review questions

Title Ref
Are Alice, Bob and Charlie telling the truth, or lying? R5157
Can we estimate a difference of eighth powers? R6503
Can we find $F(1) + F(2) + F(3) + \dots + F(100)$? R5875
Can we find $\sum_{r=1}^n 2/(r^2+4r+3)$? R5245
Can we find a good rational approximation for $\sqrt{5}$? R6340
Can we find a sequence of solutions to the equation $x^2 - 2y^2 = 1?$ R8033
Can we find an approximation to $(1.0099)^6$? R7489
Can we find an approximation to $\sqrt{5}+\sqrt{3}$? R7477
Can we find an inequality from the expansion of $[ 1+1/\sqrt{n}]^n$? R6770
Can we find how many boys study French, Latin and German? R6707
Can we find the coefficient of $x^2$ in $(5 + 4x)(1 - x/2)^{10}$? R7078
Can we find the coefficients of $x^{−12}$ and $x^2$ in this expansion? R9207
Can we prove that $\sum_{i=1}^n H_i=n^3?$ R9884
Can we prove these four binomial coefficient identities? R6102
Can we reach our target pattern in less than five moves? R9059
Can we show $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-2}{r-1}+\binom{n-3}{r-1}+\dotsb+\binom{r-1}{r-1}$? R8014
Can we show $g(n)$ is equal to the recursion depth of $f(n)$? R7507
Can we show that all words consisting of $A$s and $B$s can be made? R9750
Can we sum $r(r+1)(r+3)$ for $r$ from $1$ to $n$? R5246
Can we use a binomial expansion to evaluate $\left(19\tfrac{3}{4}\right)^6$? R7978
Can we visit every square in the grid? R7397
Does Alice like wine or beer? R6093
Given the binomial expansion of $(1+x)^n$, can we find $x$ and $n$? R6295
Given the digits $1$ to $5$, how many odd numbers can we make? R9323
Given this definition of $F(n)$, can we find the value of $F(6000)$? R8168
Given this set of tiles, when can we create a match? R6970
How many numbers $n$ satisfy $f(n)=16$? R7906
How many rows can we make with these black and white pebbles? R9664
How many students took French, Latin and German? R5848
How many ways to arrange the saints and liars are there? R6463
How many ways to choose the group to play tennis are there? R8021
If $f(n) > k$ for all $n \geq 1$, what can we say about $k$? R7031
If $x - 1/x = u$, what's $x^3 - 1/x^3$? R9928
In how many ways can we arrange $4$ white, $3$ black, and $2$ red marbles? R9341
In how many ways can we organise a mixed doubles tennis match? R9749
The liar, the switcher and the truth-teller; who's who? R8279
What do we get if we add the digits of the integers from 1 to 99? R7399
What is the coefficient of $x^3y^5$ in the expansion of $(1+xy+y^2)^n$? R5965
What is the largest value of $n$ for which $x_n>x_{n+1}$? R9473
What's the highest power of $x$ in this nested polynomial? R8605
When is $6 \times 7 = 42$ a counter-example? R6993
When will $N$ definitely be a square number? R9800
Where will this robot be after $n$ iterations? R8203
Which dates contain no repetitions of a digit? R5212
Which of Alf, Beth and Gemma is the liar? R8691
Which power of $x$ has the greatest coefficient? R5563
Who broke the vase? R5344
Who should win the game HT-2? R7878
Who will win, Amy or Brian? R7059
Who's wearing a red hat and who's wearing a blue? R5512
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