Question

Show that if \(p\) is a positive integer, then \(\left(\begin{array}{l}p \\\ n\end{array}\right)=0\) when \(n>p,\) so \((1+x)^{p}=\sum\left(\begin{array}{l}p \\\ n\end{array}\right) x^{n}\) is just a sum of \(p+1\) terms, from \(n=0\) to \(n=p .\) For example, \((1+x)^{2}\) has 3 terms, \((1+x)^{3}\) has 4 terms, etc. This is just the familiar binomial theorem.

Short answer

If \( n > p \), \ (\binom{p}{n} = 0 \. Thus, \( (1 + x)^p \) only has \ p + 1 \ terms from \( n = 0 \) to \( n = p \).

Step-by-step solution

Understand the Binomial Coefficient

The binomial coefficient \(\binom{p}{n}\) is defined as \(\frac{p!}{n!(p-n)!}\). This coefficient represents the number of ways to choose \ n \ elements from a set of \ p \ elements.

Condition When \( n > p \)

When \( n \) is greater than \( p \), the term \( (p - n)! \) in the denominator becomes negative, leading to an undefined factorial. Specifically, for \( n > p \), \( \binom{p}{n} = \frac{p!}{n!(p-n)!} = 0 \). Hence, this term does not contribute to the sum.

Formulating the Binomial Expansion

For binomials, we use \( (1 + x)^p = \binom{p}{0} x^0 + \binom{p}{1} x^1 + \binom{p}{2} x^2 + \ \dots \ \binom{p}{p} x^p \). Given from Step 2, when \( n > p \), \(\binom{p}{n}\) becomes 0, so the series stops at \( n = p \). Therefore, this expansion contains \ p+1 \ terms (from \( n = 0 \) to \( n = p \)).

Conclude with Examples

To illustrate, \( (1 + x)^2 = \binom{2}{0} x^0 + \binom{2}{1} x^1 + \binom{2}{2} x^2 \), and includes 3 terms. Similarly, \( (1 + x)^3 = \binom{3}{0} x^0 + \binom{3}{1} x^1 + \binom{3}{2} x^2 + \binom{3}{3} x^3 \) includes 4 terms.

Key concepts

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics. It is denoted as \(\binom{p}{n}\) and is defined mathematically as: \(\binom{p}{n} = \frac{p!}{n!(p-n)!}\). This represents the number of ways to choose \(\text{n}\) elements from a set of \(\text{p}\) elements.
Factors in the formula include factorials. A factorial, denoted by \(!\), is the product of all positive integers less than or equal to that number. For instance:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(3! = 3 \times 2 \times 1 = 6\)

It's important to note that when \(\text{n} > \text{p}\), the term \( (p - n)! \) becomes negative. This results in the binomial coefficient being zero because factorials for negative values are undefined. Hence, \(\binom{p}{n} = 0\) if \(\text{n} > \text{p}\).
Understanding this property is crucial, especially when dealing with binomial expansions.

Binomial Expansion

The binomial theorem allows us to expand expressions of the form \((1 + x)^p\) into a sum of terms involving binomial coefficients. The general formula is given by: \((1 + x)^p = \binom{p}{0} x^0 + \binom{p}{1} x^1 + \binom{p}{2} x^2 + ... + \binom{p}{p} x^p\).
This formula signifies that each term in the expansion is a product of a binomial coefficient and a power of \(x\). For example:

• For \((1 + x)^2\), the expanded form is \(\binom{2}{0} x^0 + \binom{2}{1} x^1 + \binom{2}{2} x^2 = 1 + 2x + x^2\)
• For \((1 + x)^3\), the expanded form is \(\binom{3}{0} x^0 + \binom{3}{1} x^1 + \binom{3}{2} x^2 + \binom{3}{3} x^3 = 1 + 3x + 3x^2 + x^3\)

From the previous section, we know that the binomial coefficient \( \binom{p}{n} = 0 \) when \( n > p \). Hence, the series has exactly \(\text{p} + 1\) terms, starting from \(\text{n} = 0\) to \(\text{n} = p\). This is crucial for simplifying calculations and identifying patterns in series.

Factorial

Factorials play an essential role in mathematical calculations and combinatorics. Factorials are denoted by an exclamation mark, like \( n! \), and are defined as the product of all positive integers less than or equal to \(n\). Here are some key points about factorials:

• \(0! = 1\) by definition.
• \(n! = n \times (n-1) \times (n-2) \times ... \times 1\)
• Factorials grow very quickly. For instance, \(5! = 120\) and \(10! = 3,628,800\).

In the context of the binomial coefficient, factorials help determine the different ways to choose subsets of elements. The formula for the binomial coefficient involves dividing the factorial of \(p\) by the product of factorials of \(n\) and \(p-n\). Factorials also appear in probability and statistics, permutations, and many fields of mathematical study.