Question:
The coefficients of $x^p$ and $x^q$ (p and q are positive integers) in the expansion of $(1+ x )^{p+q}$ are
The coefficients of $x^p$ and $x^q$ (p and q are positive integers) in the expansion of $(1+ x )^{p+q}$ are
Updated On: Jul 5, 2022
- equal
- equal with opposite signs
- reciprocals of each other
- none of these
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The Correct Option is A
Solution and Explanation
We have $t_{p + 1} = {^{p + q}C_p}\, x^p$ and $t_{q + 1} = {^{p+q}C_q} \, x^q \, {^{p + q}C_p }= {^{p + q}C_q}$. [ Remember $^nC_r = {^nC_{n- r}}$ ]
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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.