Considering the word "BUNDESLIGA," find the possible number of ways of arranging all the letters of the word and any four letters of the word. In each case, the arrangement begins with the letter D.

1. Arranging all the letters of the word "BUNDESLIGA":
Total number of letters = 10
Number of repetitions: B (2), U (1), N (1), D (1), E (1), S (1), L (1), I (1), G (1), A (1)

Number of ways = 10! / (2! * 1! * 1! * 1! * 1! * 1! * 1! * 1! * 1! * 1!) = 7257600

2. Arranging the letter D and any four other letters of the word "BUNDESLIGA":
Total number of letters = 5
Number of repetitions: B (1), U (1), N (1), E (1), S (1), L (1), I (1), G (1), A (1)

Number of ways = 5! / (1! * 1! * 1! * 1! * 1!) = 120

Therefore, there are 7257600 ways to arrange all the letters of the word "BUNDESLIGA" and 120 ways to arrange the letter D and any four other letters of the word.

Possible Arrangements Beginning with D

All Letters (10 letters):

- D + 9 other letters = 9! ways = 362,880 ways

Four Letters (4 letters):

- D + 3 other letters = 8! ways = 40,320 ways

Combinations of Four Letters:

The word "BUNDESLIGA" contains the following combinations of four letters:

- DBDL
- DBGA
- DLIG
- DLSU

Possible Arrangements for Each Combination:

- DBDL = 3! = 6 ways
- DBGA = 3! = 6 ways
- DLIG = 3! = 6 ways
- DLSU = 3! = 6 ways

Total Possible Arrangements for Four Letters Starting with D:

6 + 6 + 6 + 6 = 24 ways

Therefore, there are 24 possible ways of arranging any four letters of the word "BUNDESLIGA" in an arrangement that begins with the letter D.