AMC 10 · 2004 · #5
Grade 6 algebraIn the expression c⋅ab−d, the values of a, b, c, and d are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?
Pick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: The four numbers $0$, $1$, $2$, and $3$ are placed, one each, into $a$, $b$, $c$, and $d$ in the expression $c\cdot a^b-d$. Find the largest value the expression can reach.
Givens: The expression is $c\cdot a^b-d$.; $a$, $b$, $c$, $d$ take the values $0$, $1$, $2$, $3$ in some order, each used exactly once.; Answer choices: (A) $5$, (B) $6$, (C) $8$, (D) $9$, (E) $10$.
Unknowns: The maximum possible value of $c\cdot a^b-d$ over all ways to assign the four numbers.
Understand
Restated: The four numbers $0$, $1$, $2$, and $3$ are placed, one each, into $a$, $b$, $c$, and $d$ in the expression $c\cdot a^b-d$. Find the largest value the expression can reach.
Givens: The expression is $c\cdot a^b-d$.; $a$, $b$, $c$, $d$ take the values $0$, $1$, $2$, $3$ in some order, each used exactly once.; Answer choices: (A) $5$, (B) $6$, (C) $8$, (D) $9$, (E) $10$.
Plan
Primary tool: #14 Extreme Principle
Secondary: #2 Make a Systematic List, #3 Eliminate Possibilities
The question asks for a maximum, the signature trigger for Tool #14 (Extreme Principle): push each part of the expression toward the extreme that makes the whole biggest — subtract the smallest number you can, and build the largest product you can. Tool #2 (Make a Systematic List) keeps the few strong candidates organized so none is missed, and Tool #3 (Eliminate Possibilities) rules out the traps — wasting a big number as the multiplier, or settling for the second-best power — to confirm $9$ is truly the ceiling.
Execute — Answer: D
6.EE.A.2 Step 1 Subtract the least, build the rest
- Two moves make $c\cdot a^b-d$ as big as possible: take away as little as possible, and make the product $c\cdot a^b$ as large as possible.
- The smallest number on hand is $0$, so spend it on $d$ — subtracting $0$ removes nothing.
- This also parks the weakest number where it does no harm, since a $0$ used as a base or a multiplier would collapse the product to $0$.
💡 A result is largest when you take away the least and build everything else as high as you can.
6.EE.A.1 Step 2 The power grows fastest
- Of the three pieces, the power $a^b$ can climb the highest, so it should get the biggest numbers.
- Compare the two strong powers you can build from $\{1,2,3\}$: $3^2=9$ and $2^3=8$.
- Even though $3$ and $2$ trade places, $3^2=9$ comes out on top, so choose $a=3$ and $b=2$.
💡 Exponents pile up fast, so which number is the base and which is the power matters more than the multiply or the subtract.
6.EE.A.2 Step 3 Assign the leftovers
- Using $a=3$ and $b=2$ spends the $3$ and the $2$.
- The numbers left for $c$ and $d$ are $0$ and $1$.
- A multiplier of $0$ would wipe out the whole product, so set $c=1$ and $d=0$.
- Substituting gives $1\cdot 3^2-0=1\cdot 9-0=9$.
💡 Never multiply by $0$ when you want a big product; save the $0$ for whatever gets subtracted.
6.NS.C.7 Step 4 Check the other contenders
- Confirm nothing beats $9$.
- The only other big power is $2^3=8$, which leaves $\{0,1\}$ for $c$ and $d$: the best it gives is $1\cdot 8-0=8$.
- A large multiplier cannot rescue a small power either — $c=3$ with $2^1$ gives $3\cdot 2-0=6$, and $c=2$ with $3^1$ gives $2\cdot 3-0=6$.
- Every rival lands at $8$ or below, so the maximum is $9$, choice (D).
💡 List the handful of real contenders and pick the biggest — $9$ tops them all.
6.EE.A.2 Two moves make $c\cdot a^b-d$ as big as possible: take away as little as possibl 6.EE.A.1 Of the three pieces, the power $a^b$ can climb the highest, so it should get the 6.EE.A.2 Using $a=3$ and $b=2$ spends the $3$ and the $2$. The numbers left for $c$ and $ 6.NS.C.7 Confirm nothing beats $9$. The only other big power is $2^3=8$, which leaves ${ Review
Reasonableness: The result cannot exceed the largest product you can build, and that product tops out at $1\cdot 3^2=9$; since you then subtract $d\ge 0$, the whole expression is at most $9$. That instantly rules out $10$ (choice E) as unreachable — there is no negative number here to subtract and push the total higher. The trap answers match specific mistakes: $8$ comes from settling for the power $2^3$, and $6$ comes from wasting a big number as the multiplier on a small power. Landing exactly on the offered choice $9$ is a good sign the assignment is right.
Alternative: Brute force with a systematic list. There are only $4!=24$ orderings, and symmetry trims them fast: fix $d=0$ so nothing is subtracted and $c=1$ so the product survives, then just compare the powers $a^b$ over the remaining pair — exactly $3^2=9$ against $2^3=8$. Even a full 24-row table confirms $9$ is the ceiling.
CCSS standards used (min grade 6)
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating and comparing the candidate powers $3^2=9$ and $2^3=8$ to choose the base and exponent.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Substituting a chosen assignment of $0,1,2,3$ into $c\cdot a^b-d$ and evaluating it, e.g. $1\cdot 3^2-0=9$.)6.NS.C.7Understand ordering and absolute value of rational numbers (Ordering the candidate results $9$, $8$, and $6$ to identify the maximum.)
⭐ To make an expression as big as possible, feed the fastest-growing part — the exponent — your biggest numbers, never multiply by $0$, and save the $0$ for whatever you subtract.
⭐ To make an expression as big as possible, feed the fastest-growing part — the exponent — your biggest numbers, never multiply by $0$, and save the $0$ for whatever you subtract.
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