The decision and its conditions
A binomial coefficient counts the ways to choose which factors contribute a particular term in a product. Coefficients and exponents must be tracked separately. A constant term can occur when positive and negative powers cancel, even though neither factor is constant. Factorials count ordered arrangements; an adjacency restriction can be handled by treating a pair as one block and then accounting for its internal orders. Unordered selections need a different count, and subtracting forbidden adjacent pairs works only when each excluded pair has been counted once.
Worked example
In (x+3)³, what is the coefficient of x²? Choose which one of the three factors contributes the constant. There are three choices, each contributing the constant once. The coefficient of x² is 3×3=9.
A contrasting decision
For x≠0, what is the constant term in (x+2/x)⁴? A constant requires two factors of x and two factors of 1/x so the exponents cancel. There are six ways to choose the two reciprocal factors. Each contributes 2², giving constant term 24.
Diagnose a competing method
The first example’s tempting answer 27 follows this mistake: Report the coefficient of x rather than x². The requested x² term uses two x factors and one constant, not two constants.
Transfer before revealing
5 distinct books include A and B. In how many linear arrangements are A and B not adjacent? Pause and identify the original restrictions. Solution: All arrangements number 5!. For adjacent A and B, treat them as a block: 2×4! internal and external arrangements. Subtract to obtain 72; each adjacent arrangement has been removed exactly once.
Check the boundary of the method
Factorials count ordered arrangements; an adjacency restriction can be handled by treating a pair as one block and then accounting for its internal orders. First solve the exponent-balance condition, then multiply the coefficient contributions. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
In (x+3)³, what is the coefficient of x²?
- Choose which one of the three factors contributes the constant.
- There are three choices, each contributing the constant once.
- The coefficient of x² is 3×3=9.
Compare the other answer choices
Omit the three possible positions of the constant.
Expand (x+c)² first, then omit the product of its x² term with the final constant c.
Report the coefficient of x rather than x².
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overview