\(\sum_{\substack{i,j=0 \\ t \neq j}}^n\) \(^nC_i\ ^nC_j \)
is equal to
\(\sum_{\substack{i,j=0 \\ t \neq j}}^n\) \(^nC_i\ ^nC_j \)
is equal to
\(2^{2n \text\_2n}C_n\)
\(2^{2n-1\_2n-1}C_{n-1}\)
\(2^{2n-\frac{1}{2}}\ ^{2n}C_n\)
\(2^{n-1}+2^{2n-1}C_n\)
The Correct Option is A
Solution and Explanation
The correct answer is (A) : \(2^{2n \text\_2n}C_n\)
\(\sum_{\substack{i,j=0 \\ t \neq j}}^n\) \(^nC_i\ ^nC_j \)
\(= ∑^{n}_{i,j = 0}\) \(^nC_i\ ^nC_j - ∑^{n}_{i=j}\ ^nC_i\ ^nC_j\)
\(= ∑^{n}_{j=0}\ ^nC_i ∑^{n}_{j =0}\ ^nC_j - ∑^{n}_{ i =0}\ ^nC_i\ Ci\)
\(= 2^n.2^n-\ ^{2n}C_n\)
\(= 2^{2n\_2n}C_n\)
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For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
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).