Sure! Let's solve the problem step by step.
We need to find the coefficient of the [tex]\( t^{16} p^3 \)[/tex] term in the expansion of [tex]\( (t + 2p)^{19} \)[/tex].
### Step 1: Understand the Binomial Expansion
The binomial expansion of [tex]\( (t + 2p)^{19} \)[/tex] is given by:
[tex]\[ (t + 2p)^{19} = \sum_{k=0}^{19} \binom{19}{k} t^{19-k} (2p)^k, \][/tex]
where [tex]\( \binom{19}{k} \)[/tex] is the binomial coefficient.
### Step 2: Identify Relevant Term
We are interested in the term [tex]\( t^{16} p^3 \)[/tex]. To get [tex]\( t^{16} \)[/tex] and [tex]\( p^3 \)[/tex], we need:
[tex]\[ 19 - k = 16 \][/tex]
[tex]\[ k = 3 \][/tex]
### Step 3: Substitute [tex]\( k = 3 \)[/tex] into the Binomial Expansion
The term with [tex]\( k = 3 \)[/tex] in the expansion is:
[tex]\[ \binom{19}{3} t^{19-3} (2p)^3 \][/tex]
[tex]\[ = \binom{19}{3} t^{16} (2p)^3 \][/tex]
### Step 4: Simplify the Term
The [tex]\( (2p)^3 \)[/tex] part can be simplified as:
[tex]\[ (2p)^3 = 2^3 p^3 = 8p^3. \][/tex]
Thus, the term becomes:
[tex]\[ \binom{19}{3} t^{16} 8p^3 \][/tex]
### Step 5: Calculate the Binomial Coefficient
The binomial coefficient [tex]\( \binom{19}{3} \)[/tex] is:
[tex]\[ \binom{19}{3} = \frac{19!}{3! \cdot (19-3)!} \][/tex]
However, based on the numerical result, we know:
[tex]\[ \binom{19}{3} = 969 \][/tex]
### Step 6: Calculate the Coefficient
Now, multiply the binomial coefficient by the constant factor from [tex]\( (2p)^3 \)[/tex]:
[tex]\[ \text{Coefficient} = 969 \cdot 8 = 7752 \][/tex]
### Conclusion
Hence, the coefficient of the [tex]\( t^{16} p^3 \)[/tex] term in the expansion of [tex]\( (t + 2p)^{19} \)[/tex] is:
[tex]\[ \boxed{7752} \][/tex]
