Chapter 10 - Section 10.5 - The Binomial Theorum - Exercise Set - Page 1093: 67

a) The value of $ nCr $ for $ n=8$ and $ r=3$ is $\underline{56}$ and is verified. b) The value of $ nCr $ for $ n=7$ and $ r=2$ is $\underline{21}$ and is verified c) The value of $ nCr $ for $ n=12$ and $ r=1$ is $\underline{12}$ and is verified d)The value of $ nCr $ for $ n=11$ and $ r=1$ is $\underline{11}$ and is verified e)The value of $ nCr $ for $ n=6$ and $ r=6$ is $\underline{1}$ and is verified f)The value of $ nCr $ for $ n=15$ and $ r=2$ is $\underline{105}$ and is verified g)The value of $ nCr $ for $ n=100$ and $ r=2$ is $\underline{4,950}$ and is verified h)The value of $ nCr $ for $ n=100$ and $ r=98$ is $\underline{4,950}$ and is verified

(a) Obtain the value of $\left( \begin{align} & 8 \\ & 3 \\ \end{align} \right)$ as follows: $\begin{align} & 8\text{ }nCr\text{ 3 } \\ & \text{ 56} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 8 \\ & 3 \\ \end{align} \right)=56$ and is the same as one obtained before. (b) The value of $\left( \begin{align} & 7 \\ & 2 \\ \end{align} \right)$ is found as follows: $\begin{align} & 7\text{ }nCr\text{ 2} \\ & \text{ 21} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 7 \\ & 2 \\ \end{align} \right)=21$ and is the same as one obtained before. (c) The value of $\left( \begin{align} & 12 \\ & 1 \\ \end{align} \right)$ is found as follows: $\begin{align} & 12\text{ }nCr\text{ 1} \\ & \text{ 12} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 12 \\ & 1 \\ \end{align} \right)=12$ and is the same as one obtained before. (d) The value of $\left( \begin{align} & 11 \\ & 1 \\ \end{align} \right)$ is found as follows: $\begin{align} & \text{11 }nCr\text{ 1} \\ & \text{ 11} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 11 \\ & 1 \\ \end{align} \right)=11$ and is the same as one obtained before. (e) The value of $\left( \begin{align} & 6 \\ & 6 \\ \end{align} \right)$ is found as follows: $\begin{align} & \text{6 }nCr\text{ 6} \\ & \text{ 1} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 6 \\ & 6 \\ \end{align} \right)=1$ and is the same as one obtained before. (f) Obtain the value of $\left( \begin{align} & 15 \\ & 2 \\ \end{align} \right)$ as follows: $\begin{align} & \text{15 }nCr\text{ 2} \\ & \text{ 105} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 15 \\ & 2 \\ \end{align} \right)=105$ and is the same as one obtained before. (g) The value of $\left( \begin{align} & 100 \\ & 2 \\ \end{align} \right)$ is found as follows: $\begin{align} & \text{100 }nCr\text{ 2} \\ & \text{ 4950} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 100 \\ & 2 \\ \end{align} \right)=4,950$ and is the same as one obtained before. (h) The value of $\left( \begin{align} & 100 \\ & 98 \\ \end{align} \right)$ is found as follows: $\begin{align} & \text{100 }nCr\text{ 98} \\ & \text{ 4950} \\ \end{align}$ Thus, the value of $\left( \begin{align} & 100 \\ & 98 \\ \end{align} \right)=4,950$ and is the same as one obtained before.