23. In how many different ways can the word BANANA be read from the following table by moving from one cell to another cell with which it shares an edge? Cells may be visited more than once.
A 14
B 28
C 56
D 84
E 112
Solution
D
Case 1 The first $\mathrm{N}$ is in the 1st row, 3rd column. If the $2 \mathrm{nd} \mathrm{N}$ is in the same place, then there are $2 \times 2=4$ possibilities. If the 2 nd $\mathrm{N}$ is in the 2 nd row, 2 nd column, then there are $2 \times 4=8$ possibilities. If the 2 nd $\mathrm{N}$ is in the 3 rd row, 3rd column, then there are $1 \times 2=2$ possibilities. Altogether, there are $4+8+2=14$ possibilities.

Case 2 The first $\mathrm{N}$ is in the 2 nd row, 2nd column. If the 2 nd $\mathrm{N}$ is in the same column, then there are $4 \times 4=16$ possibilities. If the 2 nd $\mathrm{N}$ is one of the other two columns, then there are $2 \times 2=4$ possibilities. Altogether, there are $2(16+3 \times 4)=56$ possibilities.

Case 3 The first $\mathrm{N}$ is in the 1st column, 3rd row. As in the first case, 14 possibilities.
This gives a total of $56+14+14$, which is 84 possibilities.
Alternate Method
The numbers in the cells shown below give the number of sequences of the given letters that end in that particular cell. The numbers at the given stage are obtained from the previous stage by adding the numbers at the previous stage in all the cells from which the given cell can be reached in one move.
The total number of ways of reading BANANNA is the sum of the numbers in the final diagram, that is, $18+18+24+24=84$.