Question:
$ ^mC_{r+1} $ + $ \sum_{k =m}^{n} $ $ ^kC_r = $
$ ^mC_{r+1} $ + $ \sum_{k =m}^{n} $ $ ^kC_r = $
Updated On: Jul 28, 2022
- $ ^nC_{r+1} $
- $ ^{n+1} C_{r+1} $
- $ ^nC_r $
- None of these
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The Correct Option is B
Solution and Explanation
$^{m}C_{r+1}$+ $\displaystyle\sum_{k=m}^{n}$ $^{k}C_{r}$
$=\left(^{m}C_{r+1}+^{m}C_{r}\right)+^{m+1}C_{r}+^{m+2}C_{r} +\ldots^{m+n}C_{r}$
$=\left(^{m+1}C_{r+1}+^{m+1}C_{r}\right)+^{m+2}C_{r} +\cdots ^{m+n}C_{r} $
$= ^{m+2}C_{r+1} +^{m+2}C_{r}+\ldots+ ^{n}C_{r}$
$\begin{matrix}:&:&:\\ :&:&:\end{matrix} $
$=^{n+1}C_{r+1}$
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Notes on Combinations
Concepts Used:
Combinations
The method of forming subsets by selecting data from a larger set in a way that the selection order does not matter is called the combination.
- It means the combination of about ‘n’ things taken ‘k’ at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.
- For example, Imagine you go to a restaurant and order some soup.
- Five toppings can complement the soup, namely:
- croutons,
- orange zest,
- grated cheese,
- chopped herbs,
- fried noodles.
But you are only allowed to pick three.
- There can be several ways in which you can enhance your soup with savory.
- The selection of three toppings (subset) from the five toppings (larger set) is called a combination.
Use of Combinations:
It is used for a group of data (where the order of data doesn’t matter).