Question 4 A coin is flipped 10 times where each flip comes

Question 4. A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes (i) are there in total? (ii) contain exactly two heads? (iii) contain at most three tails? (iv) contain the same number of heads and tails?

Solution

i.

In each toss 2 possible outcomes so total 2^{10}=1024 outcomes

ii.

We choose 2 of the ten outcomes in C(10,2)=10*9/2=45 ways and rest other are tails

So, 45 outcomes

iii.

Case 1. 1 tail

Choose one outcome in C(10,1) =10 ways

Case 2. 2 tails

Choose 2 outcomes in C(10,2)=45 ways

Case 3. 3 tails

Choose 3 outcomes in C(10,3)=120 ways

So total

120+45+10=175 ways

iv)

So we must have 5 heads and 5 tails

Choose 5 outcomes in C(10,5) ways