Question

There are 30 marbles in a bag, 10 are green, 8 are blue, $m$ are red and $n$ are yellow, where $m$ and $n$ are integers and $m \geq n$. When two marbles are selected from the bag without replacement the probability they are the same colour is $\frac{107}{435}$. Find the value of $\left(m^2+n m+n^2\right)$.
[Note: $10,8, m$ and $n$ add to 30 .]

Answer 1

$P ($ two drawn marbles are of the same colour $)=\frac{{ }^{10} C _2+{ }^8 C _2+{ }^{ m } C _2+{ }^n C _2}{{ }^{30} C _2}$
$\therefore \frac{107}{435}=\frac{10 \cdot 9+8 \cdot 7+ m ( m -1)+ n ( n -1)}{30 \cdot 29} \Rightarrow \frac{107}{87}=\frac{90+56+\left( m ^2+ n ^2\right)-( m + n )}{6 \cdot 29} $
$\Rightarrow 214=146+\left( m ^2+ n ^2\right)+12 \Rightarrow 226-146= m ^2+ n ^2 \Rightarrow 80= m ^2+(12- m )^2 $
$\Rightarrow 2 m ^2-24 m +144=80 \Rightarrow 2 m ^2-24 m +64=0 \Rightarrow m ^2-12 m +32=0 \Rightarrow( m -8)( m -4)=0$
$\therefore m =8 \text { and } n=4 $
$\therefore m ^2+ mn + n ^2=64+32+4=100 $