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If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal to
- a)2
- b)-1
- c)1
- d)-2
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If the sum of the coefficients in the expansion of (α2x2 - 2&alp...
We need to find the sum of all coefficients in the expansion of $(a+b+c+d)^{10}$.
By the Binomial Theorem, the coefficient of $a^p b^q c^r d^s$ in the expansion of $(a+b+c+d)^{10}$ is:
$$\binom{10}{p,q,r,s} = \frac{10!}{p!q!r!s!}$$
where $p+q+r+s=10$.
To find the sum of all coefficients, we need to sum over all possible values of $p,q,r,s$ that satisfy $p+q+r+s=10$:
$$\sum_{p+q+r+s=10} \binom{10}{p,q,r,s} = \sum_{p=0}^{10}\sum_{q=0}^{10-p}\sum_{r=0}^{10-p-q}\binom{10}{p,q,r,10-p-q-r}$$
We can simplify this expression by using the Hockey Stick Identity, which states that:
$$\sum_{k=n}^{m}\binom{k}{n} = \binom{m+1}{n+1}$$
Using this identity twice, we get:
\begin{align*}
\sum_{p=0}^{10}\sum_{q=0}^{10-p}\sum_{r=0}^{10-p-q}\binom{10}{p,q,r,10-p-q-r} &= \sum_{p=0}^{10}\sum_{q=0}^{10-p}\binom{11-p-q}{2}\\
&= \sum_{p=0}^{10}\binom{12-p}{3}\\
&= \binom{13}{4}\\
&= \boxed{715}
\end{align*}
Therefore, the sum of all coefficients in the expansion of $(a+b+c+d)^{10}$ is 715.
By the Binomial Theorem, the coefficient of $a^p b^q c^r d^s$ in the expansion of $(a+b+c+d)^{10}$ is:
$$\binom{10}{p,q,r,s} = \frac{10!}{p!q!r!s!}$$
where $p+q+r+s=10$.
To find the sum of all coefficients, we need to sum over all possible values of $p,q,r,s$ that satisfy $p+q+r+s=10$:
$$\sum_{p+q+r+s=10} \binom{10}{p,q,r,s} = \sum_{p=0}^{10}\sum_{q=0}^{10-p}\sum_{r=0}^{10-p-q}\binom{10}{p,q,r,10-p-q-r}$$
We can simplify this expression by using the Hockey Stick Identity, which states that:
$$\sum_{k=n}^{m}\binom{k}{n} = \binom{m+1}{n+1}$$
Using this identity twice, we get:
\begin{align*}
\sum_{p=0}^{10}\sum_{q=0}^{10-p}\sum_{r=0}^{10-p-q}\binom{10}{p,q,r,10-p-q-r} &= \sum_{p=0}^{10}\sum_{q=0}^{10-p}\binom{11-p-q}{2}\\
&= \sum_{p=0}^{10}\binom{12-p}{3}\\
&= \binom{13}{4}\\
&= \boxed{715}
\end{align*}
Therefore, the sum of all coefficients in the expansion of $(a+b+c+d)^{10}$ is 715.
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Community Answer
If the sum of the coefficients in the expansion of (α2x2 - 2&alp...
To find the sum of coefficients in any expansion, we put all variables as 1.
sum of coefficients = (α^2 − 2α + 1)^51=0
(α^2 − 2α + 1)^51 = 0
α^2 − 2α + 1 = 0
(a-1)^2=0
a=1
sum of coefficients = (α^2 − 2α + 1)^51=0
(α^2 − 2α + 1)^51 = 0
α^2 − 2α + 1 = 0
(a-1)^2=0
a=1
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If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer?
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If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer?.
If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the sum of the coefficients in the expansion of (α2x2 - 2αx + 1)51 vanishes, then α is equal toa)2b)-1c)1d)-2Correct answer is option 'C'. Can you explain this answer?.
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