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fiasKO [112]
3 years ago
15

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.)
Mathematics
1 answer:
Lisa [10]3 years ago
3 0

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

   

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