The value of 1- 20 C 0-2 -20 C 1-30 C 1-1 -20 C 0-3 -20 C 2-30 C 2-21 -20 C 20-30 C 20 is 50 C 20-20 - 49 C q- where q is an evennumber- then q - q -5 is

∑ ^20 _r=0 (r+1) ^20C_r· ^30C_r =∑ ^20 _r=0 ^20C_r· ^30C_r+ ∑ ^20 _r=1 r·20/r ^19C_r 1· ^30C_r (1+x)^20= ^20C_0+ ^20C_1 x+ ^20C_2 x^2+… ...[1] (1+x)^3 ...

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Byju's Answer

Standard XII

Physics

Dimensional Analysis

The value of ...

Question

The value of
$1 \cdot^{20} C_{0} + 2 \cdot^{20} C_{1} \cdot^{30} C_{1} + 1 \cdot^{20} C_{0} + + 3 \cdot^{20} C_{2} \cdot^{30} C_{2} + \dots + 21 \cdot^{20} C_{20} \cdot^{30} C_{20}$ is $^{50} C_{20} + 20 \cdot^{49} C_{q}$ , (where $q$ is an even number), then $\frac{q}{q + 5}$ is

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Solution

$20 \sum r = 0 (r + 1)^{20} C_{r} \cdot^{30} C_{r} = 20 \sum r = 0^{20} C_{r} \cdot^{30} C_{r} + 20 \sum r = 1 r \cdot \frac{20}{r}^{19} C_{r - 1} \cdot^{30} C_{r}$

$(1 + x)^{20} =^{20} C_{0} +^{20} C_{1} x +^{20} C_{2} x^{2} + \dots . . . [1] (1 + x)^{30} =^{30} C_{0} x^{30} +^{30} C_{1} x^{29} +^{30} C_{2} x^{28} + \dots . . . [2]$

Coefficient of $x^{30}$ in the product of $[1]$ and $[2]$ will be
$= 20 \sum r = 0^{20} C_{r} \cdot^{30} C_{r}$

Coefficient of $x^{30}$ in $(1 + x)^{50} =^{50} C_{30}$
$∴ 20 \sum r = 0^{20} C_{r} \cdot^{30} C_{r} =^{50} C_{30}$

$(1 + x)^{19} =^{19} C_{0} +^{19} C_{1} x +^{19} C_{2} x^{2} + \dots . . . [3] (1 + x)^{30} =^{30} C_{0} x^{30} +^{30} C_{1} x^{29} +^{30} C_{2} x^{28} + \dots . . . [4]$

Coefficient of $x^{29}$ in the product of $[3]$ and $[4]$ will be
$= 20 \sum r = 1^{19} C_{r - 1} \cdot^{30} C_{r}$

Coefficient of $x^{29}$ in $(1 + x)^{49} =^{49} C_{29} =^{49} C_{20}$
$∴ 20 \sum r = 1^{19} C_{r - 1} \cdot^{30} C_{r} =^{49} C_{20}$

$20 \sum r = 0 (r + 1)^{20} C_{r} \cdot^{30} C_{r} =^{50} C_{20} + 20 \cdot^{49} C_{20}$

$∴ q = 20$

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The value of 1⋅ 20C0+2⋅ 20C1⋅ 30C1+1⋅ 20C0++3⋅ 20C2⋅ 30C2+…+21⋅ 20C20⋅ 30C20 is 50C20+20⋅ 49Cq, (where q is an even number), then qq+5 is

The value of
$1 \cdot^{20} C_{0} + 2 \cdot^{20} C_{1} \cdot^{30} C_{1} + 1 \cdot^{20} C_{0} + + 3 \cdot^{20} C_{2} \cdot^{30} C_{2} + \dots + 21 \cdot^{20} C_{20} \cdot^{30} C_{20}$ is $^{50} C_{20} + 20 \cdot^{49} C_{q}$ , (where $q$ is an even number), then $\frac{q}{q + 5}$ is