Worked solution
Classify code words by the number of repeated pairs
Using the letters of the word CORRELATION, find the number of
(i) arrangements such that the two letters A and T are next to each other,
(ii) 6-letter code words that can be formed.
[2011/AJC/II/6(modified)]
CJC · Discussion · Question 10
Answer
(i) 1814400; (ii) 114660.
Read the available multiplicities
CORRELATION has 11 letters of 9 distinct types: O and R each occur twice; C, E, L, A, T, I and N each occur once. Letters may be used only up to these available multiplicities.
(i) Group A and T
Treat A and T as one block with internal orders. There are 10 objects, with two O’s and two R’s indistinguishable. The count is .
(ii) No repeated pair
Choose and order six distinct types from 9: .
(ii) Exactly one repeated pair
Choose which type is doubled (O or R), choose four other distinct types from the remaining 8, then arrange six letters with one pair: . The other repeatable type may occur once here, but not twice.
(ii) Both repeated pairs
Use O, O, R, R and choose two of the 7 singleton types. This gives . Add the three disjoint repetition patterns: .
Common mistakes
“9 distinct letters” in the source means 9 letter types, not 9 physical letters in the word.
Solution notes
All printed asks, working, counting factors and source annotations visually compared and retained. No marks are printed for this question.