A) \[{{40.}^{69}}{{C}_{29}}\]
B) \[{{40.}^{70}}{{C}_{30}}\]
C) \[^{69}{{C}_{29}}\]
D) \[^{70}{{C}_{30}}\]
Correct Answer: A
Solution :
[a] \[\sum\limits_{r=0}^{40}{r.\frac{40}{r}{{.}^{39}}{{C}_{r-1}}{{.}^{30}}{{C}_{r}}}\] \[=40\sum\limits_{r=1}^{40}{^{39}{{C}_{r-1}}{{.}^{30}}{{C}_{r}}}\] \[{{(1+x)}^{39}}{{=}^{39}}{{C}_{0}}{{+}^{39}}{{C}_{1}}x+....{{+}^{39}}{{C}_{r}}{{x}^{r}}+....{{+}^{39}}{{C}_{39}}{{x}^{39}}\] ? (i) \[{{\left( x+1 \right)}^{30}}{{=}^{30}}Co{{X}^{30}}{{+}^{30}}{{C}_{1}}{{X}^{29}}{{+}^{30}}{{C}_{2}}{{X}^{28}}+...{{+}^{30}}{{C}_{30}}\] ... (ii) \[{{(1+x)}^{39}}{{(x+1)}^{30}}={{[}^{39}}{{C}_{0}}{{.}^{30}}{{C}_{1}}{{+}^{39}}{{C}_{1}}{{.}^{30}}{{C}_{2}}+....{{+}^{39}}{{C}_{29}}{{.}^{30}}{{C}_{30}}]\] Multiply (i) by (ii) and comparing the coeff of \[{{x}^{29}}.\] \[={{40.}^{69}}{{C}_{29}}\]
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