The value of ${}^{10}{{\text{C}}_1} + {}^{10}{C_2} + {}^{10}{C_3} + \ldots + {}^{10}{C_9}$is
$
{\text{A}}{\text{. }}{{\text{2}}^{10}} \\
{\text{B}}{\text{. }}{2^{11}} \\
{\text{C}}.{\text{ }}{2^{10}} - 2 \\
{\text{D}}.{\text{ }}{2^{10}} - 1 \\
$
Last updated date: 22nd Mar 2023
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Answer
310.2k+ views
Hint: - Here, we apply the formula of sum of binomial coefficient. Binomial coefficient is the number of ways to retrieve a set of p values from q different elements.
We will apply the formula of sum of binomial coefficient.
i.e. ${}^n{C_0} + {}^n{C_1} + {}^n{C_2} + \ldots + {}^n{C_n} = {2^n}$
As given in question
${}^{10}{C_1} + {}^{10}{C_2} + {}^{10}{C_3} + \ldots + {}^{10}{C_9}$
To use above formula we have to add and subtract ${}^{10}{{\text{C}}_0}$ and ${}^{10}{{\text{C}}_{10}}$.
Now, question becomes
${}^{10}{{\text{C}}_0} + {}^{10}{C_1} + {}^{10}{C_2} + \ldots + {}^{10}{C_{10}} - {}^{10}{C_0} - {}^{10}{C_{10}}$
$ \Rightarrow {2^{10}} - 2$
$\because {}^{10}{C_0} = {}^{10}{C_{10}} = 1$.
So option ${\text{C}}$ is the correct answer.
Note: - When you get these types of summation questions in binomial, you have to proceed as that question is set on any formula, or try to set a formula by addition or subtraction.
We will apply the formula of sum of binomial coefficient.
i.e. ${}^n{C_0} + {}^n{C_1} + {}^n{C_2} + \ldots + {}^n{C_n} = {2^n}$
As given in question
${}^{10}{C_1} + {}^{10}{C_2} + {}^{10}{C_3} + \ldots + {}^{10}{C_9}$
To use above formula we have to add and subtract ${}^{10}{{\text{C}}_0}$ and ${}^{10}{{\text{C}}_{10}}$.
Now, question becomes
${}^{10}{{\text{C}}_0} + {}^{10}{C_1} + {}^{10}{C_2} + \ldots + {}^{10}{C_{10}} - {}^{10}{C_0} - {}^{10}{C_{10}}$
$ \Rightarrow {2^{10}} - 2$
$\because {}^{10}{C_0} = {}^{10}{C_{10}} = 1$.
So option ${\text{C}}$ is the correct answer.
Note: - When you get these types of summation questions in binomial, you have to proceed as that question is set on any formula, or try to set a formula by addition or subtraction.
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