15.13. A Magic Trick 499
a,b,c,d,e,f,g, respectively. The code word 2449249 has patterncaabcab,
which is obtained by replacing its characters4,9,2bya,b,c, respectively.
(d)What length 7 code word has three occurrences of 7 , two occurrences of 8 ,
one occurrence each of 2 and 9 , and patternabacbad?
(e)Explain why the number of 9-vertex numbered trees with degree sequence
.4;3;2;2;1;1;1;1;1/is the product of the answers to parts (b) and (c).
Problems for Section 15.8
Class Problems
Problem 15.22.
The Tao of BOOKKEEPER: we seek enlightenment through contemplation of the
wordBOOKKEEPER.
(a)In how many ways can you arrange the letters in the wordPOKE?(b)In how many ways can you arrange the letters in the wordBO 1 O 2 K? Observe
that we have subscripted the O’s to make them distinct symbols.
(c)Suppose we map arrangements of the letters inBO 1 O 2 Kto arrangements
of the letters inBOOKby erasing the subscripts. Indicate with arrows how the
arrangements on the left are mapped to the arrangements on the right.
O 2 BO 1 K
KO 2 BO 1
O 1 BO 2 K
KO 1 BO 2
BO 1 O 2 K
BO 2 O 1 K
:::
BOOK
OBOK
KOBO
:::
(d)What kind of mapping is this, young grasshopper?(e)In light of the Division Rule, how many arrangements are there ofBOOK?(f)Very good, young master! How many arrangements are there of the letters in
KE 1 E 2 PE 3 R?
(g)Suppose we map each arrangement ofKE 1 E 2 PE 3 Rto an arrangement of
KEEPERby erasing subscripts. List all the different arrangements ofKE 1 E 2 PE 3 R
that are mapped toREPEEKin this way.