How many permutations of the letters ABCDEFG contain the string BCD?

Yulia

Yulia

Answered question

2021-10-08

How many permutations of the letters ABCDEFG contain the string BCD?

Answer & Explanation

oppturf

oppturf

Skilled2021-10-09Added 94 answers

Definitions
Definition permutation (order is important):
P(n,r)=n!(nr)!
Definition combination (order is not important):
C(n,r)=(nr)=n!r!(nr)!
with n!=n(n1)...21.
We treat the mentioned string as one single letter, thus we have then the possible letters A, BCD, E,F,G.
We then have 5 letters of which we want to select 5 letters.
n=5
r=5
Evaluate the definition of a permutation:
P(5,5)=5!(55)!=5!0!=5120

karton

karton

Expert2023-05-26Added 613 answers

Step 1: Count the total number of permutations of the letters ABCDEFG.
The number of permutations of a set of n distinct objects is given by n!.
Step 2: Count the number of permutations that contain the string BCD.
To count the number of permutations that contain the string BCD, we can treat BCD as a single unit. So, we have 4 units: BCD, A, E, F, G. The number of permutations of these 4 units is 4!.
Step 3: Count the number of permutations where BCD can appear in any position.
Since BCD can appear in any position within the 4 units, we need to multiply the number of permutations in Step 2 by the number of possible positions for BCD. In this case, there are 5 possible positions: at the beginning, after A, after E, after F, and after G. Therefore, we multiply 4! by 5.
Step 4: Calculate the final result.
To find the number of permutations of the letters ABCDEFG that contain the string BCD, we multiply the result from Step 3 by the total number of permutations in Step 1.
Putting it all together:
Total number of permutations=Step 1×Step 3=7!×4!×5=(7×6×5×4×3×2×1)×(4×3×2×1)×5=5040×24×5=604,800
Therefore, there are 604,800 permutations of the letters ABCDEFG that contain the string BCD.
star233

star233

Skilled2023-05-26Added 403 answers

To solve the problem of finding how many permutations of the letters ABCDEFG contain the string BCD, we can use the concept of permutations with restrictions.
Let's denote the string BCD as a single entity, say X. Now, we need to find the number of permutations of the letters ABCDEFG where X appears together.
To find the total number of permutations, we consider X as a single letter, which means we have 6 distinct entities (A, X, E, F, G) to arrange. The total number of permutations of these entities is 6!.
However, within the entity X, there are 3 distinct letters (B, C, D), which can be arranged in 3! ways.
Therefore, the total number of permutations of the letters ABCDEFG containing the string BCD is 6!×3!.
alenahelenash

alenahelenash

Expert2023-05-26Added 556 answers

Answer:
7!(5!×3!)=7×6×5×4×3×2×15×4×3×2×1×3×2×1
Explanation:
First, let's determine the total number of permutations of the letters ABCDEFG. Since we have 7 distinct letters, the number of permutations is given by 7!.
7!=7×6×5×4×3×2×1
Next, we need to consider the number of permutations that contain the string BCD. To do this, we can treat the string BCD as a single unit.
Now, we have 5 distinct units: BCD, A, E, F, G. The number of permutations of these units is given by 5!.
5!=5×4×3×2×1
However, within the BCD unit, there are 3 distinct letters (B, C, D) that can be rearranged. So, we need to consider the number of permutations within the BCD unit, which is given by 3!.
3!=3×2×1
To find the number of permutations that contain the string BCD, we multiply the number of permutations of the remaining units (5!) by the number of permutations within the BCD unit (3!).
5!×3!=5×4×3×2×1×3×2×1
Finally, we can find the number of permutations that do not contain the string BCD by subtracting the number of permutations that contain BCD from the total number of permutations.
7!(5!×3!)=7×6×5×4×3×2×15×4×3×2×1×3×2×1

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?