ข้อ 9 Combinatorics คับ
$$A-B=(1^2-1)\binom{53}{1}+(2^2-2)\binom{53}{2}+...+(53^2-53)\binom{53}{53} $$
$$=(1-1)\frac{53!}{(0)!(52)!}+(2-1)\frac{53!}{(1)!(51)!}+...+(53-1)\frac{53!}{(52)!(0)!} $$
$$=53\left(\,(0)\binom{52}{0}+(1)\binom{52}{1}+...+(52)\binom{52}{52} \right)$$
$$=53(52)(2)^{51}=53(13)(2)^{53} \therefore k=53$$
|