이항계수 확률분포

$\displaystyle \sum\limits _{x=1} ^{6} {\mathrm{P}} ( X = x )$$\displaystyle = \frac{1}{k} \sum\limits _{x=1} ^{6} {}_{6} {\mathrm{C}} _{x}$$\displaystyle = \frac{1}{k} ( 2 ^{6} - 1 )$$= 1$이므로 $k = 2 ^{6} - 1$$= 63$$= 3 ^{2} \times 7$ ${\mathrm{E}} ( X )$$\displaystyle = \sum\limits _{x=1} ^{6} x \cdot {\mathrm{P}} ( X = x )$ $\displaystyle = \frac{1}{k} \sum\limits _{x=1} ^{6} x \cdot _{6} {\mathrm{C}} _{x}$$\displaystyle = \frac{2 ^{6}}{k} \sum\limits _{x=1} ^{6} x \cdot _{6} {\mathrm{C}} _{x} \left( \frac{1}{2} \right) ^{6}$$\displaystyle = \frac{2 ^{6}}{k} \sum\limits _{x=0} ^{6} x \cdot _{6} {\mathrm{C}} _{x} \left( \frac{1}{2} \right) ^{6}$ $\displaystyle = \frac{2 ^{6}}{k} \cdot 3$ $\displaystyle \left( \begin{array}{l} \begin{matrix} \\ \end{matrix} \end{array} \right.$∵ 이항분포 $\displaystyle {\mathrm{B}} \left( 6 , \frac{1}{2} \right)$를 따르는 확률변수의 확률질량함수는 $\displaystyle {}_{6} {\mathrm{C}} _{x} \left( \frac{1}{2} \right) ^{6}$이고 평균은 $\displaystyle \sum\limits _{x=0} ^{6} x \cdot _{6} {\mathrm{C}} _{x} \left( \frac{1}{2} \right) ^{6}$$\displaystyle = 6 \times \frac{1}{2} = 3$$\displaystyle \begin{matrix} \\ \end{matrix} )$ 따라서 $\displaystyle m = \frac{2 ^{6}}{k} \cdot 3$에서 $k ^{2} m = 2 ^{6} \cdot 3 k = 2 ^{6} \cdot 3 ^{3} \cdot 7$ ∴ $a = 6$, $b = 3$, $c = 1$ ∴ $a + b + c = 10$ [다른 풀이] 항등식 $\displaystyle ( x + 1 ) ^{n} = \sum\limits _{i=0} ^{n} {}_{n} {\mathrm{C}} _{i} x ^{i}$의 양변을 $x$에 대하여 미분하면 $\displaystyle n ( x + 1 ) ^{n-1} = \sum\limits _{i=1} ^{n} i \times _{n} {\mathrm{C}} _{i} x ^{i-1}$ 위 식의 양변에 $x$$= 1$을 대입하면 $n \cdot 2 ^{n-1}$$\displaystyle = \sum\limits _{i=1} ^{n} i \times _{n} {\mathrm{C}} _{i}$ ∴ $\displaystyle \sum\limits _{i=1} ^{6} x \times _{6} {\mathrm{C}} _{x}$$= 6 \cdot 2 ^{5} = 3 \cdot 2 ^{6}$