Question:
After simplification, what is the number of terms in the expansion of $[(3x + y)^5]^4 - [(3x-y)^4]^5$ ?
After simplification, what is the number of terms in the expansion of $[(3x + y)^5]^4 - [(3x-y)^4]^5$ ?
Updated On: Jul 6, 2022
- 4
- 5
- 10
- 11
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The Correct Option is C
Solution and Explanation
Given expression is :
$[(3x + y)^5]^4 - [(3x - y)^4]^5 = [(3x + y)]^{20} - [(3x - y)]^{20}$
First and second expansion will have 21 terms each but odd terms in second expansion
be Ist, 3rd, 5th,.....21st will be equal and opposite to those of first expansion.
Thus, the number of terms in the expansion of above exapression is 10.
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.