An urn contains marbles of four colours: red, | vedclass

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(B) Let the number of red,white,blue,and green marbles be $r, w, b, g$ respectively,and $r + w + b + g = n$.Given that the events are equally likely,w

Vedclass · Accepted Answer

$21$

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(B) Let the number of red,white,blue,and green marbles be $r, w, b, g$ respectively,and $r + w + b + g = n$.Given that the events are equally likely,we have:$\frac{{}^rC_4}{{}^nC_4} = \frac{{}^wC_1 \cdot {}^rC_3}{{}^nC_4} = \frac{{}^wC_1 \cdot {}^bC_1 \cdot {}^rC_2}{{}^nC_4} = \frac{{}^rC_1 \cdot {}^wC_1 \cdot {}^bC_1 \cdot {}^gC_1}{{}^nC_4}$From the first equality: ${}^rC_4 = {}^wC_1 \cdot {}^rC_3$ $\Rightarrow \frac{r(r-1)(r-2)(r-3)}{24} = w \cdot \frac{r(r-1)(r-2)}{6}$ $\Rightarrow r-3 = 4w$ $\Rightarrow r = 4w + 3$.From the second equality: ${}^wC_1 \cdot {}^rC_3 = {}^wC_1 \cdot {}^bC_1 \cdot {}^rC_2$ $\Rightarrow \frac{r-2}{3} = b$ $\Rightarrow r = 3b + 2$.From the third equality: ${}^wC_1 \cdot {}^bC_1 \cdot {}^rC_2 = {}^rC_1 \cdot {}^wC_1 \cdot {}^bC_1 \cdot {}^gC_1$ $\Rightarrow {}^rC_2 = r \cdot g$ $\Rightarrow \frac{r(r-1)}{2} = r \cdot g$ $\Rightarrow r-1 = 2g$ $\Rightarrow r = 2g + 1$.Equating the expressions for $r$: $r = 4w + 3 = 3b + 2 = 2g + 1$.For $r$ to be an integer,$r \equiv 3 \pmod 4$,$r \equiv 2 \pmod 3$,and $r \equiv 1 \pmod 2$.Solving these congruences,$r \equiv 11 \pmod{12}$.For the smallest positive integer $r$,$r = 11$.Then $w = (11-3)/4 = 2$,$b = (11-2)/3 = 3$,and $g = (11-1)/2 = 5$.Total marbles $n = r + w + b + g = 11 + 2 + 3 + 5 = 21$.

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