Chapter 11.9, Problem 42PPSE

Solution

To calculate: The value of $8_{C_4}$ and the row and term in Pascal’s triangle and use combinations to find the 8^th term of the 13^th row’s value.

The value of $8_{C_4}$ is 70 and it is 4^th terms of the 8^th row in Pascal’s triangle. The value of 8^th term of the 13^th row value is ${13}_{C_8}=1287$

Given:

The term $8_{C_4}$ and 8^th term of the 13^th row.

Concept used:In Pascal’s triangle, nth row has n+1 terms each term represents the combinations of $n_{C_0}$ to $n_{C_n}$ . The formula used to calculate combinations is $n_{C_r}=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

From the given information the value of $8_{C_4}$ is the 4^th value of the 8^th row. In Pascal’s triangle counting of rows and values starts from 0. With reference to below shown Pascal’s triangle

  

The $8_{C_4}$ is the 4^th term of the 8^th row, with reference to Pascal’s triangle its value is 70.

The 8^th term of the 13^th row is ${13}_{C_8}$ and we get the following using the combinations formula

  $\begin{array}{l}{n}_{C_r}=\frac{n!}{r!\left(n-r\right)!}\\ {13}_{C_8}=\frac{13!}{8!\left(13-8\right)!}=\frac{62270202800}{40320\times 120}=1287\end{array}$

The value of 8^th term in the 13^th row is 1287.