Respuesta :
We can evaluate case by case from 5 tails to 7 tails. As both cases are included we have:
5 tails out of 8
6 tails out of 8
7 tails out of 8
The number of ways we can have 5 tails trying 8 times is just 8 combined 5. We use combination since the order does not matter here.
Recalling the formula for combination:
[tex]nCk=\frac{n!}{k!(n-k)!}[/tex]Then:
[tex]8C5=\frac{8!}{5!(8-5)!}[/tex]Solving:
[tex]8C5=\frac{8!}{5!(8-5)!}=\frac{8\cdot7\cdot6\cdot(5\cdot4\cdot3\cdot2\cdot1)}{(5\cdot4\cdot3\cdot2\cdot1)\cdot(3)!}[/tex]We can cancel the factors in the parentheses of numerator and denominator since they are the same:
[tex]8C5=\frac{8\cdot7\cdot6}{3!}=\frac{8\cdot7\cdot6}{3\cdot2\cdot1}=\frac{8\cdot7\cdot6}{6}=8\cdot7=56[/tex]The first combination is 56 then.
When tossing a coin 8 times, we have 56 different ways to obtain 5 tails.
Now we need to do the same process for 6 and 7 tails and add all 3 results at the end.
Calculating the number of possible ways to obtain 6 tails out of 8:
[tex]8C6=\frac{8!}{6!\cdot2!}=\frac{8\cdot7\cdot6!}{6!\cdot2!}=\frac{8\cdot7}{2}=28[/tex]When tossing a coin 8 times, we have 28 different ways to obtain 6 tails.
Now for 7 tiles:
[tex]8C7=\frac{8!}{7!\cdot1!}=\frac{8\cdot7!}{7!}=8[/tex]When tossing a coin 8 times, we have 8 different ways to obtain 7 tails.
Finally, the number of ways a person can toss a coin 8 times and obtain between 5 and 7 tails, inclusive is:
[tex]56+28+8=92\text{ ways}[/tex]Then, the answer is 92 ways.
