If 30C12+2 30C22+3 30C32+……+30 30C302=α 60 ! (30 ! )2 then α is equal to

30C12+2 30C22+3 30C32+….+30 30C302=α⋅⌊60(∣30_)2Wkt a⋅C02+(a+d)C12+(a+2d)C22+…..+(a+nd)Cn2=(2a+nd)2⋅2nCnNow 0⋅ 30C02+1⋅ 30C12+230C22+……..+30⋅ 30C302−0⋅ 30C02=α⋅60_(30_)2⇒2(0)+30(1)260C30−0=α⋅ 60C30 Here a=0,d=1⇒15⋅60C30=α⋅60C30⇒α=15

If 30C12+2 30C22+3 30C32+……+30 30C302=α 60 ! (30 ! )2 then α is equal to

30C12+2 30C22+3 30C32+….+30 30C302=α⋅⌊60(∣30_)2Wkt a⋅C02+(a+d)C12+(a+2d)C22+…..+(a+nd)Cn2=(2a+nd)2⋅2nCnNow 0⋅ 30C02+1⋅ 30C12+230C22+……..+30⋅ 30C302−0⋅ 30C02=α⋅60_(30_)2⇒2(0)+30(1)260C30−0=α⋅ 60C30 Here a=0,d=1⇒15⋅60C30=α⋅60C30⇒α=15

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If ${(^{30} C_{1})}^{2} + 2 {(^{30} C_{2})}^{2} + 3 {(^{30} C_{3})}^{2} + \dots \dots + 30 {(^{30} C_{30})}^{2} = \frac{α 60!}{(30!)^{2}} then α$ is equal to

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Detailed Solution

${(^{30} C_{1})}^{2} + 2 {(^{30} C_{2})}^{2} + 3 {(^{30} C_{3})}^{2} + \dots . + 30 {(^{30} C_{30})}^{2} = α \cdot \frac{⌊ 60}{(\underline{∣ 30})^{2}}$
Wkt $a \cdot C_{0}^{2} + (a + d) C_{1}^{2} + (a + 2 d) C_{2}^{2} + \dots . . + (a + nd) C_{n}^{2} = \frac{(2 a + nd)}{2} \cdot^{2 n} C_{n}$
Now ${0 \cdot {(^{30} C_{0})}^{2} + 1 \cdot {(^{30} C_{1})}^{2} + 2^{30} C_{2})}^{2} + \dots \dots . . + 30 \cdot {(^{30} C_{30})}^{2} - 0 \cdot {(^{30} C_{0})}^{2} = α \cdot \frac{\underline{60}}{(\underline{30})^{2}}$
$\begin{array}{l} \Rightarrow {(\frac{2 (0) + 30 (1)}{2})}^{60} C_{30} - 0 = α \cdot (^{60} C_{30}) Here a = 0, d = 1 \\ \Rightarrow 15 \cdot^{60} C_{30} = α \cdot^{60} C_{30} \\ \Rightarrow α = 15 \end{array}$