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Question: What is the \[{{r}^{th}}\] term in the expansion of a binomial \[{{(x+y)}^{n}}\]? A). \[^{n}{{C}_{...

What is the rth{{r}^{th}} term in the expansion of a binomial (x+y)n{{(x+y)}^{n}}?
A). nCr−1.xn−r+1.yr−1^{n}{{C}_{r-1}}.{{x}^{n-r+1}}.{{y}^{r-1}}
B). nCr.xn−1.yr^{n}{{C}_{r}}.{{x}^{n-1}}.{{y}^{r}}
C). nCr.xn−r.yr^{n}{{C}_{r}}.{{x}^{n-r}}.{{y}^{r}}
D). nCr+1.xn−1.yr−1^{n}{{C}_{r+1}}.{{x}^{n-1}}.{{y}^{r-1}}

Explanation

Solution

First write down the binomial expression and then write its expansion. The expansion should at least contain 2-3 terms from the beginning and 2-3 terms from the end. Check out the pattern of the progressing terms and then write the general formula for (r+1)th{{(r+1)}^{th}}term to find the rth{{r}^{th}} term we have to substitute the r=r−1r=r-1 in the formula for general term we get the answer.

Complete step-by-step solution:
According to the question expression is given that is (x+y)n{{(x+y)}^{n}}
Here x, y are the real numbers and n is the positive integer.
For general information,
(x+y)n{{(x+y)}^{n}} When expanded we get:
⇒(x+y)n=nC0+nC1xn−1.y1+nC2xn−2.y2+......+nCnyn\Rightarrow {{(x+y)}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{x}^{n-1}}.{{y}^{1}}{{+}^{n}}{{C}_{2}}{{x}^{n-2}}.{{y}^{2}}+......{{+}^{n}}{{C}_{n}}{{y}^{n}}
Where nCr=n!r!(n−r)!^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}
If you observe the series then you can notice the every terms follows a pattern which is,
Power of ‘x’ keeps on consecutively decreasing, whereas that of ‘y’ increases progressively.
But, we have to find the rth{{r}^{th}}term, for that first we have to write the general term for (r+1)th{{(r+1)}^{th}}term that means we have to write the general term forTr+1{{T}_{r+1}}.
Formula for Tr+1=nCrxn−ryr{{T}_{r+1}}{{=}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}
We can see that the above general term is Tr+1{{T}_{r+1}}that is (r+1)th{{(r+1)}^{th}} term. To find out therth{{r}^{th}} term we need to replace rrin general term as r−1r-1 that means substitute r=r−1r=r-1in the general term we get:
T(r−1)+1=nCr−1xn−(r−1)y(r−1){{T}_{(r-1)+1}}{{=}^{n}}{{C}_{r-1}}{{x}^{n-(r-1)}}{{y}^{(r-1)}}
After simplifying this term we get:
Tr=nCr−1xn−(r−1)y(r−1){{T}_{r}}{{=}^{n}}{{C}_{r-1}}{{x}^{n-(r-1)}}{{y}^{(r-1)}}
Further solving this we get:
Tr=nCr−1xn−r+1yr−1{{T}_{r}}{{=}^{n}}{{C}_{r-1}}{{x}^{n-r+1}}{{y}^{r-1}}
So, the correct option is “option A”.

Note: Whenever a binomial expression is always written its expansion and also write the general term that is Tr+1{{T}_{r+1}}. If we have to find the rth{{r}^{th}} then we have to substitute rrin general term as r−1r-1to get the rth{{r}^{th}} term.
If not it would go wrong while using formula for general term because you will get the general term for (r+1)th{{(r+1)}^{th}} not rth{{r}^{th}}term.