Question

What is the \[{{r}^{th}}\] term in the expansion of a binomial \[{{(x+y)}^{n}}\]?
A). \[^{n}{{C}_{r-1}}.{{x}^{n-r+1}}.{{y}^{r-1}}\]
B). \[^{n}{{C}_{r}}.{{x}^{n-1}}.{{y}^{r}}\]
C). \[^{n}{{C}_{r}}.{{x}^{n-r}}.{{y}^{r}}\]
D). \[^{n}{{C}_{r+1}}.{{x}^{n-1}}.{{y}^{r-1}}\]

Answer 1

Hint: 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 \[{{(r+1)}^{th}}\]term to find the \[{{r}^{th}}\] term we have to substitute the \[r=r-1\] in the formula for general term we get the answer.

Complete step-by-step solution:
According to the question expression is given that is \[{{(x+y)}^{n}}\]
Here x, y are the real numbers and n is the positive integer.
For general information,
\[{{(x+y)}^{n}}\] When expanded we get:
\[\Rightarrow {{(x+y)}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{x}^{n-1}}.{{y}^{1}}{{+}^{n}}{{C}_{2}}{{x}^{n-2}}.{{y}^{2}}+......{{+}^{n}}{{C}_{n}}{{y}^{n}}\]
Where \[^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\]
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 \[{{r}^{th}}\]term, for that first we have to write the general term for \[{{(r+1)}^{th}}\]term that means we have to write the general term for\[{{T}_{r+1}}\].
Formula for \[{{T}_{r+1}}{{=}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}\]
We can see that the above general term is \[{{T}_{r+1}}\]that is \[{{(r+1)}^{th}}\] term. To find out the\[{{r}^{th}}\] term we need to replace \[r\]in general term as \[r-1\] that means substitute \[r=r-1\]in the general term we get:
\[{{T}_{(r-1)+1}}{{=}^{n}}{{C}_{r-1}}{{x}^{n-(r-1)}}{{y}^{(r-1)}}\]
After simplifying this term we get:
\[{{T}_{r}}{{=}^{n}}{{C}_{r-1}}{{x}^{n-(r-1)}}{{y}^{(r-1)}}\]
Further solving this we get:
\[{{T}_{r}}{{=}^{n}}{{C}_{r-1}}{{x}^{n-r+1}}{{y}^{r-1}}\]
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 \[{{T}_{r+1}}\]. If we have to find the \[{{r}^{th}}\] then we have to substitute \[r\]in general term as \[r-1\]to get the \[{{r}^{th}}\] term.
If not it would go wrong while using formula for general term because you will get the general term for \[{{(r+1)}^{th}}\] not \[{{r}^{th}}\]term.