Worked solution
Arrange adjacent letters and form CELEBRATE codewords
(a) Find the number of different arrangements of the nine letters in CELEBRATE in which A and T are next to each other.
(b) Find the number of 3-letter codewords that can be formed from the letters of CELEBRATE.
[2008/IJC/II/6]
CJC · Lecture · Example 14
(a) Treat A and T as a block, with two internal orders
There is one each of C,L,B,R,A,T and three E copies. Group A,T as one outer object; the 8 objects are C,E,E,E,L,B,R,[AT]. Arrange them in 8!/3! ways, then arrange A,T as AT or TA in 2! ways. The count is .
(b) Case 1: three E copies
The only codeword is EEE, contributing 1.
(b) Case 2: two E copies and one other letter
Choose 1 of the 6 other letters C,L,B,R,A,T, then arrange the resulting three letters in 3!/2! ways. The count is .
(b) Case 3: three different letter types
Choose 3 of the 7 types C,E,L,B,R,A,T, then order them in 3! ways. The count is . The source caption beneath 7 C3 says select 1 letter; the factor selects 3 different types.
(b) Add the disjoint cases
Total number of codewords=1+18+210=229. Order matters for codewords, while the available multiplicities still limit repetition: only E can appear twice or three times. No case permits repeating a single-copy letter.
Common mistakes
Use only the copies available in CELEBRATE. In(a), count both AT and TA; in(b), distinguish unordered selection of types from ordering the resulting codeword.