Understanding the Binomial Expansion and Coefficients

The question asks us to consider the binomial expansion of $\left(1 + x\right)^n$ and find the value of an expression involving the coefficients of specific terms. The binomial theorem states that the expansion of $\left(1 + x\right)^n$ is given by:

\(\left(1 + x\right)^n = \binom{n}{0} + \binom{n}{1}x + \binom{n}{2}x^2 + \dots + \binom{n}{k}x^k + \dots + \binom{n}{n}x^n\)

The coefficient of the $\left(k+1\right)$-th term in this expansion is given by $\binom{n}{k}$.

Identifying the Coefficients p, q, r, and s

We are given that p, q, r, and s are the coefficients of the first, second, n-th, and $\left(n+1\right)$-th terms respectively. Let's identify the corresponding values of k for each term:

• First term: This is the $\left(1\right)$-th term, so $k+1 = 1$, which means $k=0$. The coefficient is $p = \binom{n}{0}$.
• Second term: This is the $\left(2\right)$-nd term, so $k+1 = 2$, which means $k=1$. The coefficient is $q = \binom{n}{1}$.
• n-th term: This is the $\left(n\right)$-th term, so $k+1 = n$, which means $k=n-1$. The coefficient is $r = \binom{n}{n-1}$.
• $\left(n+1\right)$-th term: This is the $\left(n+1\right)$-th term, so $k+1 = n+1$, which means $k=n$. The coefficient is $s = \binom{n}{n}$.

Calculating the Values of p, q, r, and s

Now let's calculate the specific values of these binomial coefficients:

• $p = \binom{n}{0}$: By definition, $\binom{n}{0} = 1$. So, $p = 1$.
• $q = \binom{n}{1}$: By definition, $\binom{n}{1} = n$. So, $q = n$.
• $r = \binom{n}{n-1}$: Using the property $\binom{n}{k} = \binom{n}{n-k}$, we have $\binom{n}{n-1} = \binom{n}{n - (n-1)} = \binom{n}{1} = n$. So, $r = n$.
• $s = \binom{n}{n}$: By definition, $\binom{n}{n} = 1$. So, $s = 1$.

We can summarize the coefficients in a table:

Evaluating the Expression (ps + qr)

We need to find the value of the expression $\left(ps + qr\right)$. We substitute the values we found for p, q, r, and s:

• $p = 1$
• $q = n$
• $r = n$
• $s = 1$

Substitute these values into the expression:

\(ps + qr = \left(1\right)\left(1\right) + \left(n\right)\left(n\right)\)

Now, perform the multiplication:

\(ps + qr = 1 + n^2\)

Conclusion

The value of $\left(ps + qr\right)$ is $1 + n^2$. We compare this with the given options.

The expression $\left(ps + qr\right)$ is equal to $1 + n^2$.

Revision Table: Binomial Coefficients

Additional Information: Binomial Theorem and Properties

The binomial theorem is a fundamental concept in algebra that provides a formula for expanding powers of binomials. For $\left(1+x\right)^n$, the coefficients are the binomial coefficients $\binom{n}{k}$, which are the numbers in Pascal's triangle.

• The sum of the coefficients in the expansion of $\left(1+x\right)^n$ is obtained by setting $x=1$, which gives $\left(1+1\right)^n = 2^n$. This means $\sum_{k=0}^n \binom{n}{k} = 2^n$.
• The coefficients are symmetric: $\binom{n}{k} = \binom{n}{n-k}$. This property was used to find that $\binom{n}{n-1} = \binom{n}{1} = n$.
• The terms in the expansion are arranged in increasing powers of $x$. The first term has $x^0$, the second term has $x^1$, and the $\left(k+1\right)$-th term has $x^k$.
• The total number of terms in the expansion of $\left(1+x\right)^n$ is $n+1$. This is why the last term is the $\left(n+1\right)$-th term, corresponding to $k=n$.

Understanding these properties helps in quickly identifying the coefficients of specific terms in a binomial expansion.