Question

The value of ${1.}^{20}{C}_0{.}^{30}{C}_0+{2.}^{20}{C}_1{.}^{30}{C}_1+...+{21}^{20}{C}_{20}{.}^{30}{C}_{20}$ is

• A. ${}^{50}{C}_{20}+20{(}^{49}{C}_{30})$
• B. ${}^{50}{C}_{29}+20{(}^{49}{C}_{29})$
• C. ${}^{50}{C}_{30}+20{(}^{49}{C}_{29})$
• D. ${}^{50}{C}_{31}+20{(}^{49}{C}_{29})$

Answer 1

Answer: (C)

${}^{50}{C}_{30}+20{(}^{49}{C}_{29})$

${(1+x)}^{20}{=}_0^{20}C{+}_1^{20}C.x{+}_2^{20}C.x^2+...{+}_{20}^{20}C.x^{20}$

Multiplying both sides by $x$:
${(1+x)}^{20}.x{=}_0^{20}C.x{+}_1^{20}C.x^2{+}_2^{20}C.x^3+...{+}_{20}^{20}C.x^{21}$

Differentiating both sides:
${(1+x)}^{20}+20.x.{(1+x)}^{19}{=}_0^{20}C{+}_1^{20}C.2.x{+}_2^{20}C.3.x^2+...{+}_{20}^{20}C.21.x^{20}$ ............(i)

Now, ${(x+1)}^{30}{=}_0^{30}C.x^{30}{+}_1^{30}C.x^{29}{+}_2^{30}C.x^{28}+...{+}_{20}^{30}C$ ..............(ii)

Multiply (i) and (ii) and compare coefficients of $x^{30}$:
${(1+x)}^{50}+20.x.{(1+x)}^{49}$

$=({}_0^{30}C.x^{30}{+}_1^{30}C.x^{29}{+}_2^{30}C.x^{28}+...{+}_{20}^{30}C).({}_0^{20}C{+}_1^{20}C.2.x{+}_2^{20}C.3.x^2+...{+}_{20}^{20}C.21.x^{20})$

${\Rightarrow }_{30}^{50}C+{20.}_{29}^{49}C{=}_0^{20}C{.}_0^{30}C+{2.}_1^{20}C{.}_1^{30}C+{3.}_2^{20}C{.}_2^{30}C+...+{21.}_{20}^{20}C{.}_{20}^{30}C$