Year 11 Maths Extension 1 Module 4 Checkpoint 1

Checkpoint Quiz 1

This checkpoint covers Pascal's Triangle & Its Properties and the Binomial Theorem, Expansion (Lessons 1–2).

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Quick Review

Pascal's triangle structure: Row $n$ contains $^nC_0, \,^nC_1, \ldots, \,^nC_n$, that is $n+1$ entries. Both the row number and the position within a row are numbered starting from $0$.
Row sum property: $\displaystyle\sum_{k=0}^{n}\,^nC_k = 2^n$. Found by setting $a = b = 1$ in the Binomial Theorem, no need to add every entry individually.
Symmetry property: $^nC_k = \,^nC_{n-k}$, so every row of Pascal's triangle is a palindrome (reads the same forwards and backwards).
Hockey-stick identity: $^rC_r + \,^{r+1}C_r + \cdots + \,^nC_r = \,^{n+1}C_{r+1}$. Summing down a diagonal gives the entry one row below and one position to the right.
The Binomial Theorem: $(a+b)^n = \displaystyle\sum_{r=0}^{n}\,^nC_r\, a^{n-r} b^r$, an expansion with $n+1$ terms whose coefficients are exactly row $n$ of Pascal's triangle. Keep the sign of a negative $b$ inside every power, and bracket any coefficient on $a$ before raising it to a power.

5 random questions from the Checkpoint 1 bank, feedback shown immediately

Great work finishing Checkpoint 1.

If you struggled with any questions, go back to Lessons 1–2 to review Pascal's triangle and the Binomial Theorem before moving on.

Tick when you're ready to continue to Lesson 3.