AMC 10 · 2016 · #1
Grade 6 arithmeticPick an answer.
Computing 11!, 10!, and 9! separately means multiplying huge numbers, which is slow and error-prone. Tool #7 (Identify Subproblems) breaks the work into safe pieces: rewrite each factorial so the shared 9! shows up, factor that 9! out of the top, then cancel it with the bottom. Tool #5 (Look for a Pattern) supplies the key observation that each factorial is just a few extra factors on top of 9!, so nothing ever has to be fully multiplied out.
Expose the shared 9!
Both terms carry the same factorial.
A factorial is just the next factorial with a couple of extra factors stacked in front, so 9! is hiding inside both.
5.NBT.B.5Look For A PatternFactor 9! out of the top
Factoring it out leaves a plain difference.
If two amounts are both built from 9!, their difference is also built from 9! — just count how many 9!s are left.
If two amounts are both built from the same factorial, their difference is built from it too.
▸ Why?
A factor shared by both terms can be lifted out of the subtraction, leaving a small difference inside.
▸ Why?
Dividing by that shared factor then undoes the multiplication, so the giant piece disappears.
Cancel and read the answer
Cancelling gives 100, choice (B).
Dividing by 9! undoes multiplying by 9!, so the giant factor disappears and only 100 remains.
4.NF.A.1Identify SubproblemsWhen factorials are subtracted or divided, find the biggest one they all share, pull it out, and watch it cancel — you almost never multiply the giant numbers.
- Expose the shared 9!
- Factor 9! out of the top
- Cancel and read the answer