Question:
The value of $-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots $
$-15 .{ }^{15} C _{15}+{ }^{14} C _{1}+{ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ is
The value of $-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots $
$-15 .{ }^{15} C _{15}+{ }^{14} C _{1}+{ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ is
Updated On: Feb 14, 2025
- $2^{16}-1$
- $2^{13}-14$
- $2^{14}$
- $2^{13}-13$
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The Correct Option is B
Solution and Explanation
$\left(-{ }^{15} C _{1}+2{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots -15 .{ }^{15} C _{15}\right)$
$+\left({ }^{14} C _{1}+{ }^{14} C _{3}+\ldots .+{ }^{14} C _{11}\right)$
$=\displaystyle\sum_{ r =1}^{15}(-1)^{ r } \cdot r ^{15} C _{ r }+\left({ }^{14} C _{1}+{ }^{14} C _{3}+\ldots+{ }^{14} C _{11}+{ }^{14} C _{13}\right)-{ }^{14} C _{3}$
$=\displaystyle\sum_{ r =1}^{15}(-1)^{ r } 15 \cdot{ }^{14} C _{ r -1}+2^{13}-14$
$=15\left(-{ }^{14} C _{0}+{ }^{14} C _{1} \ldots-{ }^{14} C _{14}\right)+2^{13}-14$
$=2^{13}-14$
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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).