Answer:
1.1296
2.570
3.[tex]\frac{6}{91}[/tex]
Step-by-step explanation:
We are given that
Number of males=7 including George
Number of females=6 including Margaret
Number of children=4
Number of male selecting for roles=3
Number of females selecting for roles=1
Number of child selecting for roles=2
1.We have to find the number of ways can these roles be filled from these auditioners.
Total number of ways=[tex]7C_2\times 6C_1\times 4C_2[/tex]
[tex]nC_r=\frac{n!}{r!(n-r)!}[/tex]
Using this formula
Total number of ways=[tex]\frac{9\times 8\times 7!}{2!7!}\times \frac{6\times 5!}{1!5!}\times \frac{4\times 3\times 2!}{2!\times 2\times 1}[/tex]
Total number of ways=1296
2.We have to find number of ways can these roles be filled if exactly one of George and Margaret gets a part.
If George gets a part then Margaret out
Total number of ways=[tex]6C_2\times 5C_1\times 4C_2=\frac{6\times 5\times 4!}{2\times 1\cdot4!}\times 5\times \frac{4\times 3\times 2!}{2\times 1\cdot 2!}=450[/tex]
If Margaret gets a part then George out
Number of ways=[tex]6C_3\times 4C_2=\frac{6\times 5\times 4\times 3!}{3\times 2\times 1\times 3!}\times \frac{4\times 3\times 2}{2!\times 2\times 1}=120[/tex]
Therefore, total number of ways can these roles be filled if exactly one of George and Margaret gets a part=450+120=570
3.We have to find the probability of both George and Margaret getting a part.
Total number of audition=7+6+4=17
Except George and Margaret , number of auditions=15
Number of males=6
Probability=[tex]\frac{number\;of\;favorable\;cases}{total\;number\;cases}[/tex]
The probability of both George and Margaret getting a part=[tex]\frac{6C_2\times 4C_2}{15C_4}=\frac{\frac{6!}{2!4!}\times \frac{4!}{2!2!}}{\frac{15!}{4!11!}}[/tex]
The probability of both George and Margaret getting a part=[tex]\frac{6\times 5\times 4!}{2\times 1\times 4!}\times \frac{4\times 3\times 2!}{2!\times 2\times 1}\times \frac{4\times 3\times 2\times 11!}{15\times 14\times 13\times 12\times 11!}[/tex]
The probability of both George and Margaret getting a part=[tex]\frac{6}{91}[/tex]
