Final answer:
The question covers combinatorics, specifically calculating distinguishable permutations of the letters in the word COMBINATORICS. Adjustments are made for repeated letters and restrictions on letter pairing. Permutations are calculated using factorials and accounting for specific letter pair constraints.
Step-by-step explanation:
The question involves permutations and combinations, a topic within mathematics. To find the number of distinguishable ways to arrange the letters in COMBINATORICS, one would calculate the factorial of the number of letters, adjusting for repeated letters. However, this is not as simple as using the total number of letters due to the repetition of the letters 'C' and 'O'. For the second question, we need to calculate arrangements with the restriction that 'OO' cannot be adjacent. The third question asks for arrangements where the pair 'AB' is adjacent. Whereas the fourth question wants us to consider the 'CO' pairs occurring together.
To answer these questions systematically:
- To count the number of distinguishable permutations for COMBINATORICS, we divide 14! by the factorial of the number of times each letter repeats; here it is 2! for both 'C' and 'O' because these letters appear twice.
- To prevent 'OO' from appearing together, we can first consider 'OO' as a single element, calculate the permutations, and then subtract this from the total permutations.
- For the pair 'AB' to appear together, we treat 'AB' as a single element and then calculate the permutations of the resulting string.
- For 'CO' to appear in pairs, we group each 'CO' and calculate the permutations like in previous parts but with these 'CO' pairs considered as single elements