Question:
The value of $^{10}C_1+^{10}C_2+^{10}C_3+....+^{10}C_9 $ is
The value of $^{10}C_1+^{10}C_2+^{10}C_3+....+^{10}C_9 $ is
Updated On: May 30, 2022
- $2^{10}$
- $2^{11}$
- $2^{10}-2$
- $2^{10} -1$
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The Correct Option is C
Solution and Explanation
Since, $\left(1+x\right)^{n }= ^{n}C_{0}+^{n}C_{1}x+^{n}C_{2} x^{2} +...+^{n}C_{n} x^{n}$
Put x = 1 and n = 10, we get
$2^{10} = ^{10}C_{0}+^{10}C_{1}+^{10}C_{2}+...+^{10}C_{10}$
$2^{10} = 1+^{10}C_{1}+^{10}C_{2} +...+^{10}C_{9}+1$
$2^{10} - 2 = ^{10}C_{1}+^{10}C_{2} +...+^{10}C_{9}$
Put x = 1 and n = 10, we get
$2^{10} = ^{10}C_{0}+^{10}C_{1}+^{10}C_{2}+...+^{10}C_{10}$
$2^{10} = 1+^{10}C_{1}+^{10}C_{2} +...+^{10}C_{9}+1$
$2^{10} - 2 = ^{10}C_{1}+^{10}C_{2} +...+^{10}C_{9}$
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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).