AMC 10 · 2024 · #17
Grade 11 probabilityalgebraPick an answer.
Testing all 420 pairs (a,b) one at a time is hopeless by hand, so Tool #11 (Work Backwards) turns the problem around: instead of choosing coefficients and hunting for roots, choose the roots first and let them tell you the coefficients. Tool #4 (Introduce a Variable) makes that turn possible — name the roots p,q,r, write the cubic as (x-p)(x-q)(x-r), and expand. The constant term then forces pqr=-6, which is a tiny condition, so Tool #2 (Make a Systematic List) can write down every triple of distinct integers with product -6. Finally Tool #3 (Eliminate Possibilities) throws out the triples whose resulting a or b escapes the pool [-10,10].
Count every equally likely pair
Ordered draws give 420 pairs.
Two slots that mean different things must be filled in order, so count arrangements, not pairs.
11.S-CP.B.9Make A Systematic ListName the roots and factor
Write it as a product of three factors.
A monic polynomial is completely determined by its roots, so the roots hold all the information.
11.A-APR.B.3Introduce A VariableExpand and reverse the arrow
The roots must multiply to negative six.
Matching coefficients converts a search over 420 pairs into a search over factorizations of one small number.
Matching coefficients converts a search over many pairs into a search over factorizations of one small number.
▸ Why?
Two expressions equal for every input must agree coefficient by coefficient.
▸ Why?
Those coefficients are exactly the sums and the product of the roots, so the roots carry all the information.
List every triple with product -6
There are five distinct triples.
Because 6 has so few factorizations, the whole problem collapses to five candidates.
4.OA.B.4Make A Systematic ListRead off each pair
Each triple fixes one a and one b.
Each triple of roots produces exactly one coefficient pair, so counting triples counts pairs.
7.NS.A.2Work BackwardsDrop the pairs the pool forbids
Removing the out-of-range one leaves four.
A root triple is only useful if the coefficients it produces are numbers you were allowed to draw.
6.NS.C.7Eliminate PossibilitiesDivide favorable by total
Four over 420 is 1 over 105.
Every outcome is equally likely, so the probability is just a count divided by a count.
11.S-CP.B.9Make A Systematic ListWhen the coefficients are hard to search, build the polynomial from its roots instead, and let the constant term tell you which roots are even possible.
- Count every equally likely pair
- Name the roots and factor
- Expand and reverse the arrow
- List every triple with product -6
- Read off each pair (a,b)
- Drop the pairs the pool forbids
- Divide favorable by total