Final answer:
The number of distinguishable code symbols that can be formed with the letters of a word can be found by calculating the factorial of the number of letters in the word.
Step-by-step explanation:
a) For the word MATH, the number of distinguishable code symbols can be found by calculating the factorial of the number of letters in the word. In this case, 4! (four-factorial) is equal to 4 x 3 x 2 x 1 = 24. Therefore, there are 24 distinguishable code symbols that can be formed with the letters of the word MATH.
b) For the word BUSINESS, the number of distinguishable code symbols can be found by calculating the factorial of the number of letters in the word. In this case, 8! (eight-factorial) is equal to 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320. Therefore, there are 40,320 distinguishable code symbols that can be formed with the letters of the word BUSINESS.
c) For the word PHILOSOPHICAL, the number of distinguishable code symbols can be found by calculating the factorial of the number of letters in the word. In this case, 14! (fourteen-factorial) is equal to 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 87,792,960. Therefore, there are 87,792,960 distinguishable code symbols that can be formed with the letters of the word PHILOSOPHICAL.
d) For the word ORANGE, the number of distinguishable code symbols can be found by calculating the factorial of the number of letters in the word. In this case, 6! (six-factorial) is equal to 6 x 5 x 4 x 3 x 2 x 1 = 720. Therefore, there are 720 distinguishable code symbols that can be formed with the letters of the word ORANGE.
e) For the word BIOLOGY, the number of distinguishable code symbols can be found by calculating the factorial of the number of letters in the word. In this case, 7! (seven-factorial) is equal to 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040. Therefore, there are 5,040 distinguishable code symbols that can be formed with the letters of the word BIOLOGY.
f) For the word MATHEMATICS, the number of distinguishable code symbols can be found by calculating the factorial of the number of letters in the word. In this case, 11! (eleven-factorial) is equal to 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 39,916,800. Therefore, there are 39,916,800 distinguishable code symbols that can be formed with the letters of the word MATHEMATICS.