There are 6 positive and 8 negative numbers. From these four numbers are chosen at random and multiplied.…
There are 6 positive and 8 negative numbers. From these four numbers are chosen at random and multiplied. Then the probability, that the product is a negative number, is
$\frac {496}{1001}$
$\frac {505}{1001}$
$\frac {490}{1001}$
$\frac {504}{1001}$
Solution
Total number of numbers $=8+6=14$ 4 numbers can be chosen out of 14 numbers in ${ }^{14} \mathrm{C}_4$ ways
The product of 4 numbers will be negative, if i. one is negative and three are positive
OR
ii. three are negative and one is positive.
$\begin{aligned}
\therefore \quad \text { Required probability } & =\frac{\left({ }^8 \mathrm{C}_1 \times{ }^6 \mathrm{C}_3\right)+\left({ }^8 \mathrm{C}_3 \times{ }^6 \mathrm{C}_1\right)}{{ }^{14} \mathrm{C}_4} \\
& =\frac{(8 \times 20)+(56 \times 6)}{1001} \\
& =\frac{160+336}{1001}=\frac{496}{1001}
\end{aligned}$