For the natural numbers


For the natural numbers $\mathrm{m}, \mathrm{n}$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $\mathrm{a}_{1}=\mathrm{a}_{2}=10$, then the value of $(\mathrm{m}+\mathrm{n})$ is equal to :

  1. 88

  2. 64

  3. 100

  4. 80

Correct Option: , 4



Coefficient of y $\left(a_{1}\right)=1 .{ }^{n} C_{1}+{ }^{m} C_{1}(-1)$


Coefficient of $\mathrm{y}^{2}\left(\mathrm{a}_{2}\right)$

$=1 \cdot{ }^{\mathrm{n}} \mathrm{C}_{2}-{ }^{\mathrm{m}} \mathrm{C}_{1} \cdot{ }^{\mathrm{n}} \mathrm{C}_{1} \cdot+1 \cdot{ }^{\mathrm{m}} \mathrm{C}_{2}=10$

$=\frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{m} \cdot \mathrm{n}+\frac{\mathrm{m}(\mathrm{m}-1)}{2}=10$

$m^{2}+n^{2}-2 m n-(n+m)=20$



By equation (1) \& (2)

$\mathrm{m}=35, \mathrm{n}=45$

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