Question

suppose 6 fair coins are tossed. use pascals triangle to find the number of ways of obtaining exactly 5 heads. there are different ways of obtaining exactly 5 heads.

Explanation:

Step1: Recall Pascal's triangle property

The $n$-th row of Pascal's triangle gives the coefficients of the binomial expansion $(a + b)^n$. When tossing $n$ coins, the number of ways to get $k$ heads is given by the $(k + 1)$-th element in the $n$-th row of Pascal's triangle. Here $n=6$ and $k = 5$.

Step2: Identify the row and position in Pascal's triangle

The first few rows of Pascal's triangle are:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
The 6 -th row (counting the first row as row 0) of Pascal's triangle is 1 6 15 20 15 6 1. The number of ways to get $k$ heads in $n$ coin - tosses is given by the binomial coefficient $\binom{n}{k}$, which is the $(k + 1)$-th entry in the $n$-th row. For $n = 6$ and $k=5$, we look at the 6 -th entry (since $k + 1=6$) of the 6 -th row.

Answer:

6