If 
$ 2^m 3^n 5^k, \text{ where } m, n, k \in \mathbb{N}, \text{ then } m + n + k \text{ is equal to:} $
If
$ 2^m 3^n 5^k, \text{ where } m, n, k \in \mathbb{N}, \text{ then } m + n + k \text{ is equal to:} $
Show Hint
- 19
- 21
- 18
- 20
The Correct Option is A
Approach Solution - 1
Thus, \( m + n + k = 17 + 1 + 1 = 19 \).
Thus, the correct answer is \( 19 \).
Approach Solution -2
Step 1: The given summation is:
\[ \sum_{r=1}^{15} r^2 \binom{15}{r} \Rightarrow 15 \sum_{r=1}^{15} r^{14} \binom{r-1}{1} \]
Step 2: Simplifying the summation:
\[ 15 \sum_{r=1}^{15} (r - 1 + 1)^{14} \binom{r-1}{1} \] which further simplifies to: \[ 15 \cdot 14 \sum_{r=1}^{15} \binom{r-2}{13} + 15 \cdot 14 \sum_{r=1}^{15} \binom{r-1}{14}. \]
Step 3: Calculating the terms:
\[ 15 \cdot 14 \cdot 2^{13} + 15 \cdot 14 \cdot 2^{14}. \]
Step 4: Final Simplification:
This can be simplified as: \[ 3^1 \cdot 2^{13} (70 + 10) = 3^1 \cdot 2^{13} \cdot 80. \]
Step 5: Final Calculation:
Further simplifying: \[ 3^1 \cdot 5^1 \cdot 2^{17}. \]
Step 6: Final Result:
\[ m = 17n = 1k = 1. \]
\(m+n+k = 17+1+1 = 19\)
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