Combinatorics question about the four-letter sequence "GRIT"

In summary, combinatorics is a branch of mathematics that deals with counting, arranging, and selecting objects or events in a systematic way. A four-letter sequence is a sequence of four letters that can be arranged in different ways to form new words or strings. There are 24 possible four-letter sequences that can be formed using the letters "GRIT", and the letters in a four-letter sequence can be repeated. Combinatorics has many practical applications in various fields such as computer science, economics, and genetics.
  • #1
RM86Z
23
6
Homework Statement
Given the letters in "FULBRIGHT" how many contain the four-letter sequence "GRIT".
Relevant Equations
6 x 5!
Question: "A total of 9! = 362880 different nine-letter ‘‘words’’ can be produced by rearranging the letters in FULBRIGHT. Of these, how many contain the four-letter sequence GRIT?"

Solution: There are six ways of getting the word "GRIT" with five letters left over giving 6 x 5! = 720 possibilities.

There is no answer in my book so just wanted to verify whether my solution is correct or not.
 
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  • #2
Yes, you can consider GRIT as a single letter X. So ##6!##
 
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  • #3
Thank you I didn't think of it that way but that does make sense!
 

FAQ: Combinatorics question about the four-letter sequence "GRIT"

What is the total number of possible combinations for the four-letter sequence "GRIT"?

The total number of possible combinations for the four-letter sequence "GRIT" is 24. This can be calculated by taking the factorial of the number of letters (4!) which equals 24.

How many different arrangements can be made with the letters in "GRIT"?

There are 24 different arrangements that can be made with the letters in "GRIT". This can be calculated by taking the factorial of the number of letters (4!) which equals 24.

What is the probability of getting the sequence "GRIT" in a random arrangement of the letters?

The probability of getting the sequence "GRIT" in a random arrangement of the letters is 1 in 24, or approximately 4.17%. This can be calculated by taking the number of possible combinations (24) and dividing it by the total number of arrangements (24).

How many ways can the letters in "GRIT" be arranged if repetition is not allowed?

If repetition is not allowed, there are 12 different ways the letters in "GRIT" can be arranged. This can be calculated by taking the factorial of the number of letters (4!) and dividing it by the factorial of the number of repeated letters (2!) which equals 12.

How does the number of possible combinations change if we add an additional letter to the sequence "GRIT"?

If an additional letter is added to the sequence "GRIT", the total number of possible combinations will increase. For example, if we add the letter "E" to the sequence, the total number of combinations would become 120 (5!). This is because each additional letter adds another factorial to the calculation.

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