# 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

•

Total views: 310.2k

•

Views today: 4.87k

Answer

Verified

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.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE