Question:

The value of $ \left( \frac{^{50}{{C}_{0}}}{1}+\frac{^{50}{{C}_{2}}}{3}+\frac{^{50}{{C}_{4}}}{5}+.... \right. $ $ \left. +\frac{^{50}{{C}_{50}}}{51} \right) $ is

Updated On: Jun 7, 2024
  • $ \frac{{{2}^{50}}}{51} $
  • $ \frac{{{2}^{50}}-1}{51} $
  • $ \frac{{{2}^{50}}-1}{50} $
  • $ \frac{{{2}^{51}}-1}{51} $
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The Correct Option is A

Solution and Explanation

$ \left( \frac{^{50}{{C}_{0}}}{1}+\frac{^{50}{{C}_{2}}}{3}+\frac{^{50}{{C}_{4}}}{5}+....+\frac{^{50}{{C}_{50}}}{51} \right) $
$=\frac{1}{1}+\frac{50\times 49}{3\times 2!}+\frac{50\times 49\times 48\times 47}{5\times 4!}+.... $
$=\frac{1}{51}\left( 51+\frac{51\times 50\times 49}{3!}+\frac{\begin{align} & 51\times 50\times 49 \\ & \times 48\times 47 \\ \end{align}}{5!}+.... \right) $
$=\frac{1}{51}{{(}^{51}}{{C}_{1}}{{+}^{51}}{{C}_{3}}{{+}^{51}}{{C}_{5}}+....) $
$=\frac{1}{51}{{.2}^{51-1}}=\frac{{{2}^{50}}}{51} $
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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).