Understanding Binomial Expansion Coefficients

The question asks us to find the relationship between the coefficient of \(a^m\) and the coefficient of \(a^n\) in the binomial expansion of \((1 + a)^{m + n}\). Let's call the coefficient of \(a^m\) as \(\alpha\) and the coefficient of \(a^n\) as \(\beta\).

Applying the Binomial Theorem

The binomial theorem states that the expansion of \((x + y)^N\) is given by:

\( (x + y)^N = \sum_{k=0}^{N} \binom{N}{k} x^{N-k} y^k \)

A special case of this theorem is the expansion of \((1 + a)^N\), which is:

\( (1 + a)^N = \sum_{k=0}^{N} \binom{N}{k} 1^{N-k} a^k = \sum_{k=0}^{N} \binom{N}{k} a^k \)

In this expansion, the term containing \(a^k\) is \(\binom{N}{k} a^k\), and its coefficient is \(\binom{N}{k}\).

Finding Coefficients Alpha and Beta

In our problem, the expression is \((1 + a)^{m + n}\). Here, the total power is \(N = m + n\). The variable is \(a\).

The coefficient of \(a^m\) is \(\alpha\). Comparing with the general form, the power of \(a\) is \(m\), which means \(k = m\). So, the coefficient is \(\binom{m+n}{m}\).

\( \alpha = \text{Coefficient of } a^m \text{ in } (1 + a)^{m+n} = \binom{m+n}{m} \)

The coefficient of \(a^n\) is \(\beta\). Comparing with the general form, the power of \(a\) is \(n\), which means \(k = n\). So, the coefficient is \(\binom{m+n}{n}\).

\( \beta = \text{Coefficient of } a^n \text{ in } (1 + a)^{m+n} = \binom{m+n}{n} \)

Relationship between Binomial Coefficients

We need to find the relationship between \(\alpha\) and \(\beta\). We have:

• \(\alpha = \binom{m+n}{m}\)
• \(\beta = \binom{m+n}{n}\)

A key property of binomial coefficients is \(\binom{N}{k} = \binom{N}{N-k}\). This property arises from the definition of combinations:

\( \binom{N}{k} = \frac{N!}{k!(N-k)!} \)

and

\( \binom{N}{N-k} = \frac{N!}{(N-k)!(N-(N-k))!} = \frac{N!}{(N-k)!k!} \)

Thus, \(\binom{N}{k}\) is indeed equal to \(\binom{N}{N-k}\).

Applying the Property to Alpha and Beta

Let's apply this property to \(\alpha = \binom{m+n}{m}\). Here, \(N = m+n\) and \(k = m\).

Using the property \(\binom{N}{k} = \binom{N}{N-k}\):

\( \binom{m+n}{m} = \binom{m+n}{(m+n)-m} = \binom{m+n}{n} \)

We know that \(\alpha = \binom{m+n}{m}\) and \(\beta = \binom{m+n}{n}\). Since \(\binom{m+n}{m} = \binom{m+n}{n}\), it follows that \(\alpha = \beta\).

Conclusion

The coefficient of \(a^m\) (\(\alpha\)) and the coefficient of \(a^n\) (\(\beta\)) in the expansion of \((1 + a)^{m + n}\) are equal.

Therefore, the correct relationship is \(\alpha = \beta\).

Revision Table: Key Concepts

Additional Information: Properties of Binomial Coefficients

Binomial coefficients, denoted by \(\binom{N}{k}\) or \(^N C_k\), represent the number of ways to choose \(k\) elements from a set of \(N\) distinct elements. They appear in the binomial expansion and have several useful properties:

• \(\binom{N}{0} = 1\)
• \(\binom{N}{N} = 1\)
• \(\binom{N}{1} = N\)
• \(\binom{N}{k} = \binom{N}{N-k}\) (Symmetry property)
• \(\binom{N}{k} + \binom{N}{k+1} = \binom{N+1}{k+1}\) (Pascal's identity)
• Sum of coefficients: \(\sum_{k=0}^N \binom{N}{k} = 2^N\) (Setting \(a=1\) in \((1+a)^N\))

The symmetry property \(\binom{N}{k} = \binom{N}{N-k}\) is directly applicable here, showing that the coefficient of \(a^k\) is the same as the coefficient of \(a^{N-k}\) in the expansion of \((1+a)^N\).