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Q: GATE $2018$ - Probability question : occurrence of green and red faces when die is rolled $7$ times
A die is colored green on $4$ sides and red on $2$ sides. if the die is rolled $7$ times. Then which of following option is correct regarding occurrence of green and red faces?
(A) $2$ green and $5$ red
(B) $3$ green and $4$ red
(C) $4$ green and $3$ red
(D) $5$ green and $...
The Game is a mental game where the objective is to avoid thinking about The Game itself. Thinking about The Game constitutes a loss, which must be announced each time it occurs. It is impossible to win most versions of The Game. Depending on the variation of The Game, the whole world, or all those aware of the game, are playing it all the time. Tactics have been developed to increase the number of people aware of The Game and thereby increase the number of losses.
Though the origins of The Game are unknown, a game featuring ironic processing was played by Leo Tolstoy in 1840.
== Gameplay ==
There...
Prove that if $R$ is a commutative ring and $N=(a_1,...,a_m)$ where each $a_i$ is nilpotent., then $N$ is a nilpotent ideal.
So we need to find a $k \in \Bbb{N}$ such that $N^k = (0)$. Working through small values $m$ and $k$, I can see that $k = 1 + \max \{a_1,...,a_m\}$ seems to work. E.g...
