TS EAMCET · Maths · Binomial Theorem
If \(\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2, \ldots, \mathrm{C}_8\) are the binomial coefficients in the expansion of \((1+x)^8\) then \(\sum_{r=1}^8 r^3 \frac{\mathrm{C}_r}{\mathrm{C}_{r-1}}=\)
- A \(540\)
- B \(336\)
- C \(105\)
- D \(270\)
Answer & Solution
Correct Answer
(A) \(540\)
Step-by-step Solution
Detailed explanation
\( \frac{\mathrm{C}_r}{\mathrm{C}_{r-1}} = \frac{8-r+1}{r} = \frac{9-r}{r} \) \( \sum_{r=1}^8 r^3 \frac{9-r}{r} = \sum_{r=1}^8 r^2 (9-r) = \sum_{r=1}^8 (9r^2 - r^3) \) \( = 9 \sum_{r=1}^8 r^2 - \sum_{r=1}^8 r^3 \)…
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