Let be roots of the equation . On throwing three dice simultaneously, sum of the numbers appearing on them is . The probability such that is (where and are co-prime) then the value is:

Question

Let $α_{1}, α_{2}, \dots, α_{8}$ be roots of the equation $1 + z + z^{2} + \dots + z^{8} = 0$ . On throwing three dice simultaneously, sum of the numbers appearing on them is $n$ . The probability such that $\sum_{i = 1}^{8} {(α_{i})}^{n} = 8$ is $\frac{a}{b}$ (where $a$ and $b$ are co-prime) then the value $| 7 a - b |$ is:

Smart Achievers · Accepted Answer

(4)

$' n'$ $should be multiple of 9 for favourable event.$

$So, x_{1} + x_{2} + x_{3} = 9 or 18$

$If x_{1} + x_{2} + x_{3} = 9, 1 \leq x_{i} \leq 6$

$No. of solution of (1) = coefficient of x^{9} in$

${(x + x^{2} + \dots + x^{6})}^{3} = x^{3} {(1 - x^{6})}^{3} {(1 - x)}^{- 3}$

$= coefficient of x^{6} in (1 - 3 x^{6}) {(1 - x)}^{- 3}$

$= {}^{8}{C_{6}} + (- 3) \times 1 = 25$

$And, there is only one solution of$ $x_{1} + x_{2} + x_{3} = 18, 1 \leq x_{i} \leq 6$

$Hence, n (E) = 26, Also n (S) = 6^{3}$

$\Rightarrow P (E) = \frac{13}{108}$

Solution

1 22

2 16

3 26

4 17

Want Us to Email you About Special Offers & Updates?