How many strings of length 12 over the alphabet {a, b, c} have exactly three a's or have exactly three b's or haveexactly three c's?O 3 () 2°O 3. () – 3· ()O 3. ()- 2° - 3 ()O 3.() - 2° – 3 · () C)

Combinatorics is the field of mathematics in which we deal with counting. Counting is based on the fundamental principle of counting. According to this principle, two works can be done together in mn different ways if one work is done in m different ways, and for each of these ways, another work can be done in n different ways. The length of the string is 12, and it is over the alphabets a,b,c. In the given question, it is asked to find the number of such strings having exactly three a's or three b's or three c's. There are 12 places in the string. It has exactly three a's. Three a's can be placed in 12 places in 123 ways. The number of remaining places is 9. Each of the remaining places can be filled with b and c. The number of such combinations is 29. Thus, by the principle of counting, 12 places can be filled together having exactly three a's is 12329. Therefore, there are 12329 strings of length 12 with exactly three a's. Similarly, there are 12329 strings of length 12 with exactly three b's and 12329 strings of length 12 with exactly three c's. Therefore, the total number of strings with exactly three a's or three b's or three c's is 312329. Hence, the correct answer is option A.