Question

example 2 - distinct objects in a fixed order
how many ways are there to arrange the letters in the word number if the consonants must remain in the original order?
what is a combination in math?
an arrangement of objects in which the order is not important is called a combination. this is different from permutation where the order matters. for suppose we are arranging the letters a, b and c. in a permutation, the arrange and acb are different. but, in a combination, the arrangements abc and acb are the same because the orderwhat is the combination formula?
the number of combinations of n things taken r at a time is written as c(n, r) or
usually referred to as
items choose r\. remember for combinations order is n

Explanation:

Step1: Identify vowels and consonants in "NUMBER"

The word "NUMBER" has 6 letters, with 2 vowels (U, E) and 4 consonants (N, M, B, R). Since consonants must remain in original order, we are essentially finding arrangements of vowels in the available positions.

Step2: Use combination - like approach

The total number of positions is 6. We need to find the number of ways to place 2 vowels in these 6 positions. The number of ways to choose 2 positions out of 6 for the vowels (when order of placement of vowels matters) can be calculated using the formula for permutations of choosing $r$ objects from $n$ positions. Here $n = 6$ and $r=2$. The formula for permutations $P(n,r)=\frac{n!}{(n - r)!}$.
\[P(6,2)=\frac{6!}{(6 - 2)!}=\frac{6!}{4!}=\frac{6\times5\times4!}{4!}=30\]

Answer:

30