Question

A city council consists of eight Democrats and eight Republicans. If a committee of six people is selected, find the probability of selecting two Democrats and four Republicans.
(Type answer a fraction Simplify your answer.)

Answer 1

Answer:

The  probability is  $P[ D n R] = 0.196$

Step-by-step explanation:

  From the question we are told that

     The number of Democrats is  $D = 8$

       The number of republicans is  $R = 8$

The  number of ways of selecting selecting two Democrats and four Republicans.

         $N = \left {D} \atop {}} \right. C_2 * \left {R} \atop {}} \right. C_1$

Where C represents combination

substituting values

           $N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1$

           $N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8!}{(8-2)! 2!} * \frac{8! }{(8-4)! 1 !}$

=>        $N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8!}{(6)! 2!} * \frac{8! }{(6)! 1 !}$

=>        $N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8 * 7 * 6!}{(6)! 2!} * \frac{8*7 *6! }{(6)! 1 !}$

=>        $N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8 * 7 }{ 2*1 } * \frac{8*7 }{ 1 *1 }$

=>      $N = 1568$

The total number of ways of selecting the committee of six people is  

          $Z = \left {D+R} \atop {}} \right. C_6$

substituting values

           $Z = \left {8+8} \atop {}} \right. C_6$

            $Z= \left {16} \atop {}} \right. C_6$

substituting values

             $Z= \left {16} \atop {}} \right. C_6 = \frac{16! }{(16-6) ! 6!}$

           $Z= \left {16} \atop {}} \right. C_6 = \frac{16 *15 *14 * 13 * 12 * 11 * 10! }{10 ! 6!}$

           $Z= \left {16} \atop {}} \right. C_6 = \frac{16 *15 *14 * 13 * 12 * 11 }{6* 5 * 4 * 3 * 2 * 1}$

           $Z= \left {16} \atop {}} \right. C_6 = 8008$

The probability of selecting two Democrats and four Republicans  is  mathematically  represented as

           $P[ D n R] = \frac{N}{Z}$

substituting values

           $P[ D n R] = \frac{1568}{8008}$

            $P[ D n R] = 0.196$