Respuesta :
Answer:
Step-by-step explanation:
GIVEN: Legislative Committee A legislative committee consists of [tex]5[/tex] Democrats and [tex]4[/tex] Republicans. A delegation of [tex]3[/tex] is to be selected to visit a small Pacific island republic.
TO FIND: (a) How many different delegations are possible? (b) How many delegations would have all Democrats? (c) How many delegations would have [tex]2[/tex] Democrats and [tex]1[/tex] Republican? (d) How many delegations would include at least [tex]1[/tex] Republican.
SOLUTION:
(a)
Total members [tex]=9[/tex]
Total delegation possible[tex]=^9C_3=\frac{9!}{6!3!}[/tex]
[tex]=\frac{9\times8\times7}{6}=84[/tex]
(b)
Total democrats [tex]=5[/tex]
Total delegation possible with all democrats [tex]=^5C_3=\frac{5!}{3!2!}[/tex]
[tex]=10[/tex]
(c)
Total democrats [tex]=4[/tex]
Total republicans [tex]=5[/tex]
Total delegations with [tex]2[/tex] democrats and [tex]1[/tex] republican [tex]=^4C_1\times^5C_2=\frac{4!}{3!1!}\times\frac{5!}{2!3!}[/tex]
[tex]=4\times10=40[/tex]
(d)
Total republicans [tex]=4[/tex]
Total delegation with at least [tex]1[/tex] republican [tex]=\text{total delegation possible}-\text{total delegation with all democrats}[/tex]
putting values from part (a) and (b)
Total delegation with at least [tex]1[/tex] republican [tex]=84-10=74[/tex]
