Question

A county commissioner will randomly select 5 people to form a non-partisan committee to look into the issue of county services. If there are 8 Democrats and 6 Republicans to choose from, what is the probability that the Democrats will have the more members than the Republicans on this committee?

• A. $\frac{686}{2002}$
• B. $\frac{1316}{2002}$
• C. $\frac{1876}{2002}$
• D. $\frac{2688}{24024}$
• E. $\frac{21336}{24024}$

Answer 1

The correct option is B $\frac{1316}{2002}$

$5$ people need to be selected from $8$ democrats and $6$ republicans.

Democrats are more than republicans

$5$ democrats, $0$ republicans $\Rightarrow {}^8{C}_5$

$4$ democrats, $1$ republicans $\Rightarrow {}^8{C}_4\times {}^6{C}_1$

$3$ democrats, $3$ republicans $\Rightarrow {}^8{C}_3\times {}^6{C}_2$$={}^8{C}_4\times {}^6{C}_1{+}^8{C}_3\times {}^6{C_2+}^8{C}_5$$=420+840+4856=1316$

Total number of ways of selecting $5$ people from $14$ people $={}^{14}{C}_5$

$=\frac{14!}{5!\times 9!}$$=2002$

probability $=\frac{1316}{2002}$