Let's find out the number of substrings that can be formed from the given string. The string is "a d e f b g h n m p".

First, we need to determine the length of the string. The string has 10 characters.

A substring is a contiguous sequence of characters within a string. For example, in the string "abc", the substrings are "a", "b", "c", "ab", "bc", "abc", and sometimes the empty string "".

The total number of possible substrings (including the empty string) for a string of length \(n\) is given by the formula:

\( \text{Number of substrings} = \frac{n(n+1)}{2} + 1 \)

If we are only considering non-empty substrings, the formula is:

\( \text{Number of non-empty substrings} = \frac{n(n+1)}{2} \)

In this question, the options suggest that the count includes the empty string, as 56 is one of the options, which results from using the formula including the empty string.

Let's apply the formula for the total number of substrings (including the empty string) to the given string "a d e f b g h n m p".

The length of the string is \(n = 10\).

Using the formula:

\( \text{Number of substrings} = \frac{10(10+1)}{2} + 1 \)

\( \text{Number of substrings} = \frac{10(11)}{2} + 1 \)

\( \text{Number of substrings} = \frac{110}{2} + 1 \)

\( \text{Number of substrings} = 55 + 1 \)

\( \text{Number of substrings} = 56 \)

So, there are 56 possible substrings that can be formed from the string "a d e f b g h n m p", including the empty substring.

Let's quickly see how substrings are formed for a smaller string, say "ab" (length n=2):

Length 1 substrings: "a", "b" (2 substrings)

Length 2 substrings: "ab" (1 substring)

Empty substring: "" (1 substring)

Total substrings = 2 + 1 + 1 = 4

Using the formula \( \frac{n(n+1)}{2} + 1 \) with n=2:

\( \frac{2(2+1)}{2} + 1 = \frac{2(3)}{2} + 1 = \frac{6}{2} + 1 = 3 + 1 = 4 \)

This confirms the formula works for total substrings including the empty string.

Let's also list them for a string of length 3, say "abc" (n=3):

Length 1: "a", "b", "c" (3)

Length 2: "ab", "bc" (2)

Length 3: "abc" (1)

Empty: "" (1)

Total = 3 + 2 + 1 + 1 = 7

Using the formula \( \frac{n(n+1)}{2} + 1 \) with n=3:

\( \frac{3(3+1)}{2} + 1 = \frac{3(4)}{2} + 1 = \frac{12}{2} + 1 = 6 + 1 = 7 \)

This further confirms the formula for total substrings.

Now, let's see how the substrings are counted based on their starting position for a string of length n=3 ("abc"):

The total number of non-empty substrings is 3 + 2 + 1 = 6, which is \( \frac{3(3+1)}{2} \). When the empty string is included, the total becomes 6 + 1 = 7.

For the string "a d e f b g h n m p" with length \(n = 10\), the number of substrings is calculated as \( \frac{10 \times 11}{2} + 1 = 55 + 1 = 56 \).

Revision Table: Understanding Substrings

This table summarizes key concepts about substrings.

Additional Information on String Concepts

Besides substrings, strings have other related concepts:

• Subsequence: A sequence that can be derived from another sequence by deleting zero or more elements without changing the order of the remaining elements. Subsequences are not necessarily contiguous. For example, "ac" is a subsequence of "abc".
• Prefix: A substring that occurs at the beginning of the string. For "abc", prefixes are "", "a", "ab", "abc".
• Suffix: A substring that occurs at the end of the string. For "abc", suffixes are "", "c", "bc", "abc".

It's important to differentiate between these concepts when dealing with string manipulation problems.