Understanding Binomial Coefficients and Their Maximum Value

The binomial coefficient, denoted as C(n, r) or $\binom{n}{r}$, represents the number of ways to choose $r$ items from a set of $n$ distinct items. For a fixed natural number $n$, the values of C(n, r) for $r = 0, 1, 2, \dots, n$ form a row in Pascal's triangle. These values increase from $r=0$ up to a certain point and then decrease towards $r=n$. The greatest value(s) of C(n, r) for a fixed $n$ occur in the middle of this sequence.

Maximum Value of C(n, r) Based on n

The position of the greatest value(s) of C(n, r) depends on whether $n$ is an even or an odd number.

Case 1: n is an even number

If $n$ is even, say $n = 2k$ for some integer $k$, the sequence of binomial coefficients C(n, 0), C(n, 1), ..., C(n, n) has a single peak. The maximum value occurs at the very middle term, which is when $r = n/2$.

For example, if $n=4$, the coefficients are C(4,0)=1, C(4,1)=4, C(4,2)=6, C(4,3)=4, C(4,4)=1. The greatest value is C(4,2)=6, which occurs at $r=2$. Here, $n=4$ is even, and $r=2 = n/2$. Notice that $n = 2r$ (4 = 2*2) holds for this maximizing $r$.

If $n=6$, the coefficients are C(6,0)=1, C(6,1)=6, C(6,2)=15, C(6,3)=20, C(6,4)=15, C(6,5)=6, C(6,6)=1. The greatest value is C(6,3)=20, which occurs at $r=3$. Here, $n=6$ is even, and $r=3 = n/2$. Notice that $n = 2r$ (6 = 2*3) holds for this maximizing $r$.

Analyzing Statement 1

Statement 1 says: C(n, r) is greatest if n = 2r

This statement suggests that for C(n, r) to be the greatest value for a fixed $n$, the condition $n = 2r$ must hold for that specific $r$. This implies $r = n/2$. As we've seen, this is where the maximum occurs when $n$ is even. However, the statement implies this is the condition for C(n,r) to be greatest in general. This is not true for odd $n$, as we will see in the next case. For example, if $n=5$, the greatest values are C(5,2) and C(5,3). For $r=2$ or $r=3$, $n=5$ is not equal to $2r$. Thus, C(n,r) can be greatest even if $n \neq 2r$. Therefore, statement 1 is not a universally correct condition for C(n,r) to be greatest.

Case 2: n is an odd number

If $n$ is odd, say $n = 2k + 1$ for some integer $k$, the sequence of binomial coefficients C(n, 0), C(n, 1), ..., C(n, n) has two equal peak values. These maximum values occur at the two middle terms, which are when $r = (n-1)/2$ and $r = (n+1)/2$. Note that $(n-1)/2 + (n+1)/2 = n$, and the two values are adjacent integers.

For example, if $n=5$, the coefficients are C(5,0)=1, C(5,1)=5, C(5,2)=10, C(5,3)=10, C(5,4)=5, C(5,5)=1. The greatest values are C(5,2)=10 and C(5,3)=10, which occur at $r=2$ and $r=3$. Here, $n=5$ is odd, $(n-1)/2 = (5-1)/2 = 2$, and $(n+1)/2 = (5+1)/2 = 3$.

Let's check the relationships between $n$ and $r$ for these maximizing values:

• For $r = (n-1)/2$: $2r = n-1$, which means $n = 2r + 1$. For $n=5, r=2$, this is $5 = 2(2)+1 = 5$.
• For $r = (n+1)/2$: $2r = n+1$, which means $n = 2r - 1$. For $n=5, r=3$, this is $5 = 2(3)-1 = 5$.

So, when $n$ is odd, the greatest values occur for the two r-values that satisfy $n = 2r + 1$ or $n = 2r - 1$.

Analyzing Statement 2

Statement 2 says: C(n, r) is greatest if n = 2r - 1 and n = 2r + 1

The phrasing "n = 2r - 1 and n = 2r + 1" is mathematically impossible for a single value of $n$ and $r$. However, based on the context of binomial coefficients and the typical way such properties are described, and considering the maximizing values occur at two different r-values when n is odd, this statement is likely intended to refer to the odd case. It suggests that the condition for C(n,r) to be greatest involves relationships $n=2r-1$ and $n=2r+1$ being satisfied by the maximizing $r$ values when $n$ is odd. As shown in the example for $n=5$, the two maximizing r values (2 and 3) indeed satisfy $n=2r+1$ (for r=2) and $n=2r-1$ (for r=3).

Interpreting statement 2 charitably, it describes the conditions on $n$ and the maximizing $r$ values in the case when $n$ is odd and has two equal greatest coefficients. This covers the situation for odd $n$ where C(n,r) is greatest.

Conclusion

Statement 1 provides a condition ($n=2r$ or $r=n/2$) which is where the maximum occurs only when $n$ is even. It is not the general condition for C(n,r) to be greatest for any $n$. Thus, statement 1 is incorrect.

Statement 2, despite its awkward "and" phrasing, describes the relationships between $n$ and the maximizing $r$ values ($n=2r-1$ and $n=2r+1$) which are satisfied by the two central terms when $n$ is odd. This statement correctly identifies the situation for maximum C(n,r) when $n$ is odd. Thus, statement 2 is correct under the intended interpretation.

Therefore, only statement 2 is correct.

Revision Table: Binomial Coefficient Maxima

Additional Information: Binomial Coefficients C(n, r)

The formula for the binomial coefficient C(n, r) is given by:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

where $n!$ (n factorial) is the product of all positive integers up to $n$.

Important properties of binomial coefficients include:

• Symmetry: C(n, r) = C(n, n-r). This shows why the sequence of values is symmetric around the middle.
• Summation: $\sum_{r=0}^{n} \binom{n}{r} = 2^n$.
• Pascal's Identity: C(n, r) + C(n, r+1) = C(n+1, r+1). This identity is the basis for constructing Pascal's triangle.

The problem focuses on finding the largest term(s) in the expansion of $(x+y)^n$, where the coefficients are C(n, r).