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Equal coefficients
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$\begin{align*} &A = \left ( p x + q \right )^{504} \text{ where p and q are +ve integers and }gcd(p,q) = 1 \\ &\text{Given } \left [ x^4 \right ] = \left [ x^5 \right ] , \text{ where } \left [ x^k \right ] \text{ denote the coefficient of }x^k \text{ in A.} \\ &\text{What is p+q ?} \end{align*}$
discrete-mathematics
combinational-circuits
asked
Jul 1, 2017
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Combinatory
dd
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4
101??
4
As, $\left [ x^{4}\right ]=\left [ x^{5} \right ]$
$\binom{504}{4}.p^{4}.q^{500}=\binom{504}{5}.p^{5}.q^{499}$
=$\binom{504}{4}.q=\binom{504}{5}.p$
$\frac{p}{q}=\frac{1}{100}$
So, p+q=101
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