Question:
If
\(\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}\)then L is equal to _____.
If
\(\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}\)then L is equal to _____.
\(\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}\)then L is equal to _____.
Updated On: Mar 20, 2025
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Correct Answer: 221
Solution and Explanation
\(\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 =1^2\ ^{10}C_1^2 + 2^2\ ^{10}C_2^2 + … + 10^2\ ^{10}C_{10}\end{array}\)
Let, \(\begin{array}{l}\left(1 + x\right)^{10} = ^{10}C_0 + ^{10}C_1 x + ^{10}C_2 x^2 + ….+ ^{10}C_{10} x^{10} \end{array}\)
\(\begin{array}{l}\Rightarrow 10\left(1 + x\right)^9 = ^{10}C_1 + 2\cdot ^{10}C_2 x +… + 10\cdot ^{10}C_{10} x^9 …\left(1\right)\end{array}\)
Similarly, \(\begin{array}{l}10\left(x + 1\right)^9 = 10\cdot ^{10}C_0 x^9 + 9\cdot ^{10}C_1 x^8 + … + 1\cdot ^{10}C_9\end{array}\)
\(\begin{array}{l}100\left(1+ x\right)^{18}\text{has required term with coefficient of} ~x^9 \end{array}\)
\(\begin{array}{l}i.e. ^{18}C_9 100 = 22000 L\\ \Rightarrow L = 221\end{array}\)
Let, \(\begin{array}{l}\left(1 + x\right)^{10} = ^{10}C_0 + ^{10}C_1 x + ^{10}C_2 x^2 + ….+ ^{10}C_{10} x^{10} \end{array}\)
\(\begin{array}{l}\Rightarrow 10\left(1 + x\right)^9 = ^{10}C_1 + 2\cdot ^{10}C_2 x +… + 10\cdot ^{10}C_{10} x^9 …\left(1\right)\end{array}\)
Similarly, \(\begin{array}{l}10\left(x + 1\right)^9 = 10\cdot ^{10}C_0 x^9 + 9\cdot ^{10}C_1 x^8 + … + 1\cdot ^{10}C_9\end{array}\)
\(\begin{array}{l}100\left(1+ x\right)^{18}\text{has required term with coefficient of} ~x^9 \end{array}\)
\(\begin{array}{l}i.e. ^{18}C_9 100 = 22000 L\\ \Rightarrow L = 221\end{array}\)
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.