The largest 5-digit number having three different digits is
98978
99897
99987
98799
Step 1: Understand the goal.
We need the largest 5-digit number that uses exactly three different digits.
Step 2: Make the leftmost digits as big as possible.
The biggest digit is 9. Put 9s in the leftmost places first:
\[ 9\_\_\_\_ \]
\[ 99\_\_\_ \]
\[ 999\_\_ \]
Step 3: We still need two other digits (different from 9).
Choose the next biggest digits: 8 and 7.
Step 4: Place the bigger of these (8) as left as possible.
Put 8 in the thousands place, and 7 in the units place:
\[ 99987 \]
Step 5: Check the conditions.
Digits used: \(\{9, 8, 7\}\) → exactly 3 different digits.
Number of digits: 5 → okay.
Step 6: Compare with other options.
\[ 99987 > 99897 \quad (\text{3rd digit } 9 > 8) \]
\[ 99987 > 98978 \quad (\text{2nd digit } 9 > 8) \]
\[ 99987 > 98799 \quad (\text{2nd digit } 9 > 8) \]
Therefore, the largest is \(99987\). (Option C)