Questions: Use the binomial formula to find the coefficient of the t^2 s^21 term in the expansion of (3t+s)^23.

Solution Steps

To find the coefficient of the \( t^2 s^{21} \) term in the expansion of \( (3t + s)^{23} \), we can use the binomial theorem. The binomial theorem states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In this case, \( a = 3t \), \( b = s \), and \( n = 23 \). We need to find the term where the power of \( t \) is 2 and the power of \( s \) is 21. This corresponds to \( n-k = 2 \) and \( k = 21 \). We can then use the binomial coefficient \( \binom{23}{21} \) and the corresponding powers of \( 3t \) and \( s \).

We need to find the coefficient of the term \( t^2 s^{21} \) in the expansion of \( (3t + s)^{23} \). According to the binomial theorem, the general term in the expansion can be expressed as:

\[ \binom{n}{k} (3t)^{n-k} s^k \]

where \( n = 23 \) and \( k \) is the exponent of \( s \).

For the term \( t^2 s^{21} \), we have:

• \( k = 21 \) (the exponent of \( s \))
• \( n - k = 2 \) (the exponent of \( t \))

We calculate the binomial coefficient \( \binom{23}{21} \):

\[ \binom{23}{21} = \binom{23}{2} = \frac{23 \times 22}{2 \times 1} = 253 \]

Now, we substitute \( n-k = 2 \) and \( k = 21 \) into the expression for the coefficient:

\[ \text{Coefficient} = \binom{23}{21} (3t)^{2} s^{21} = 253 \cdot (3^2) \cdot (1) = 253 \cdot 9 = 2277 \]

Final Answer