Use the binomial theorem to find the coefficient of x^3 in (1 + x)^5.

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

Use the binomial theorem to find the coefficient of x^3 in (1 + x)^5.

Explanation:
The binomial theorem tells us that in (1+x)^n the coefficient of x^k is C(n,k). Here n is 5 and k is 3, so the coefficient is C(5,3) = 5!/(3!2!) = 10. Expanding confirms this: (1+x)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5, so the x^3 term has coefficient 10. The other numbers in the choices relate to different powers or different expansions (for example, 5 is the coefficient of x^1, and 20 would be the coefficient of x^3 in (1+x)^6), but for x^3 in (1+x)^5 the answer is 10.

The binomial theorem tells us that in (1+x)^n the coefficient of x^k is C(n,k). Here n is 5 and k is 3, so the coefficient is C(5,3) = 5!/(3!2!) = 10. Expanding confirms this: (1+x)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5, so the x^3 term has coefficient 10. The other numbers in the choices relate to different powers or different expansions (for example, 5 is the coefficient of x^1, and 20 would be the coefficient of x^3 in (1+x)^6), but for x^3 in (1+x)^5 the answer is 10.

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