Binomial Theorem and Applications

Develop general term, selected coefficient, applied patterns.

Lesson 3 of 33 parts Practice Flashcards

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General Term

Objective: Explain general term, apply the rule, and justify relevant restrictions.

The term C(n,k)u^(n−k)v^k appears in (u+v)^n for k=0,…,n. Identify the requested power before expanding everything. It counts ways to choose the k factors contributing v. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$(x+y)ⁿ=Σₖ₌₀ⁿ C(n,k)x^(n−k)y^k$

Worked examples

Try itIn (x+3)^7, find the coefficient of x².Show answer

Step 1: Choose the two factors contributing x: C(7,2)=21. Step 2: The other 5 factors contribute 3^5. Step 3: Multiply to get 5103.

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General Term

Objective: Explain general term, apply the rule, and justify relevant restrictions.

Formulas

$(x+y)ⁿ=Σₖ₌₀ⁿ C(n,k)x^(n−k)y^k$

Worked examples

Try itIn (x+3)^7, find the coefficient of x².Show answer

Step 1: Choose the two factors contributing x: C(7,2)=21. Step 2: The other 5 factors contribute 3^5. Step 3: Multiply to get 5103.

Selected Coefficient

Objective: Explain selected coefficient, apply the rule, and justify relevant restrictions.

To find one coefficient, match the variable power to k and calculate that term; a full expansion may be unnecessary. Its power contributes an additional numeric factor. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

Coefficient of x^k in (x+a)ⁿ is C(n,k)a^(n−k)

Worked examples

Try itFind the x coefficient of (x+3)^7 for positive integer c.Show answer

Step 1: Choose one x from the 7 factors. Step 2: The other 6 factors contribute 3^6. Step 3: Multiply 7·729=5103.

Applied Patterns

Objective: Explain applied patterns, apply the rule, and justify relevant restrictions.

Binomial coefficients also count combinations, while an algebraic expansion includes powers and signs. Interpret which part a context asks for. No, event probabilities also need outcome probabilities and assumptions. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$C(n,k)=n!/[k!(n−k)!]$

Worked examples

Try itHow many ways can 2 students be selected from 9 students?Show answer

Step 1: Order does not matter, so use C(9,2). Step 2: Compute (9)(8)/2. Step 3: The value is 36.

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