Question:
${}^{21}C_1 + {}^{21}C_2 + \dots + {}^{21}C_{10} =$
${}^{21}C_1 + {}^{21}C_2 + \dots + {}^{21}C_{10} =$
Show Hint
The sum of the first half of binomial coefficients for odd $n$ is $2^{n-1}$.
Updated On: Apr 28, 2026
- $2^{20}$
- $2^{21}$
- $2^{21} - 1$
- $2^{21} - 2$
- $2^{20} - 1$
Show Solution
Verified By Collegedunia
The Correct Option is
Solution and Explanation
Step 1: Concept
Sum of all coefficients: $\sum_{r=0}^{21} {}^{21}C_r = 2^{21}$. Also, ${}^{21}C_r = {}^{21}C_{21-r}$.
Step 2: Analysis
$2^{21} = ({}^{21}C_0 + \dots + {}^{21}C_{10}) + ({}^{21}C_{11} + \dots + {}^{21}C_{21})$. Since the two halves are equal: $2 \times ({}^{21}C_0 + \dots + {}^{21}C_{10}) = 2^{21}$. ${}^{21}C_0 + {}^{21}C_1 + \dots + {}^{21}C_{10} = 2^{20}$.
Step 3: Calculation
The required sum is $({}^{21}C_0 + {}^{21}C_1 + \dots + {}^{21}C_{10}) - {}^{21}C_0$. Sum $= 2^{20} - 1$. Final Answer: (E)
Sum of all coefficients: $\sum_{r=0}^{21} {}^{21}C_r = 2^{21}$. Also, ${}^{21}C_r = {}^{21}C_{21-r}$.
Step 2: Analysis
$2^{21} = ({}^{21}C_0 + \dots + {}^{21}C_{10}) + ({}^{21}C_{11} + \dots + {}^{21}C_{21})$. Since the two halves are equal: $2 \times ({}^{21}C_0 + \dots + {}^{21}C_{10}) = 2^{21}$. ${}^{21}C_0 + {}^{21}C_1 + \dots + {}^{21}C_{10} = 2^{20}$.
Step 3: Calculation
The required sum is $({}^{21}C_0 + {}^{21}C_1 + \dots + {}^{21}C_{10}) - {}^{21}C_0$. Sum $= 2^{20} - 1$. Final Answer: (E)
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM permutations and combinations Questions
- How many four digit numbers $abcd$ exist such that $a$ is odd, $b$ is divisible by $3$, $c$ is even and $d$ is prime?
- There are $10$ persons including $3$ ladies. A committee of $4$ persons including at least one lady is to be formed. The number of ways of forming such a committee is
- If $^{n}C_{r-1}=28$, $^{n}C_{r}=56$ and $^{n}C_{r+1}=70$, then the value of $r$ is equal to
- The number of words that can be formed by using all the letters of the word $PROBLEM$ only one is
- How many numbers with no more than three digits can be formed using only the digits $1$ through $7$ with no digit used more than once in a given number?
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions