To find the number of ways to arrange the letters in the word "problem" while keeping the consonants in their original order, we first identify the consonants and vowels in the word.
The letters in "problem" are:
- Consonants: \( p, r, b, l, m \)
- Vowel: \( o, e \)
The consonants (5 total) must remain in the order \( p, r, b, l, m \). We can treat the arrangement of the vowels separately.
The total letters in "problem" are \( 7 \): \( p, r, o, b, l, e, m \).
Since the consonants must remain in their order, we can think of arranging the consonants in fixed positions while allowing the vowels to be placed in any positions. The positions for the consonants will be fixed, and it will subsequently create slots for the vowels.
We calculate the total number of arrangements of the letters in "problem" in which the consonants appear in the correct order.
- Identify positions: We have 7 letters total (5 consonants and 2 vowels). We need to choose 2 positions out of the 7 for the vowels.
The number of ways to choose 2 positions from 7 can be calculated using the binomial coefficient:
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]
- Arrangement of Vowels: Since there are 2 vowels \( o \) and \( e \), they can be arranged in these 2 chosen positions in \( 2! \) ways:
\[ 2! = 2 \]
- Total arrangements: For each selection of the vowel positions, the arrangement can occur in \( 2! = 2 \) different ways. Therefore, the total number of arrangements is:
\[ 21 \times 2 = 42 \]
Thus, the total number of ways to arrange the letters in the word "problem" while keeping the consonants in their original order is \( \boxed{42} \).
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