Permutations And Combinations

Concept: Permutations and Combinations - Selection with Conditions.

Step 1: Define the possible cases based on the given constraint. The condition states the team of 4 must include "at most one boy". This leads to two distinct cases: Case I: The team contains exactly 0 boys (which means exactly 4 girls). Case II: The team contains exactly 1 boy (which means exactly 3 girls).

Step 2: Calculate the number of ways for Case I. To select 4 girls from the available 6 girls, we use combinations: ways. After selecting the 4 members, we must choose 1 leader from these 4: ways. Total ways for Case I = .

Step 3: Calculate the number of ways for Case II. To select 3 girls from 6: ways. To select 1 boy from 4: ways. After forming the team of 4 (3 girls + 1 boy), we choose 1 leader from these 4 members: ways. Total ways for Case II = .

Step 4: Combine the results of both cases. Since Case I and Case II are mutually exclusive, we add their respective total ways to find the final answer.

Step 5: Calculate the final sum. Total number of ways = . $ $

We are asked to find the number of four-letter words that can be formed using the letters from the word "BARRACK". The word "BARRACK" consists of the letters: B, A, R, R, A, C, K. Thus, we have the following available letters: - B (1 time), A (2 times), R (2 times), C (1 time), K (1 time). We can form four-letter words by selecting four letters from the available ones. There are different cases depending on how many repeated letters we use. Case 1: No repeated letters. We can select 4 letters from the 5 distinct letters {B, A, R, C, K}. The number of ways to choose 4 letters from these 5 is given by: Then, we can arrange these 4 letters in any order, so the total number of words in this case is: Case 2: One letter repeated twice and two other distinct letters. We can select one letter to be repeated from {A, R}. There are 2 ways to choose which letter will be repeated. Then, we select 2 more letters from the remaining 4 distinct letters. The number of ways to do this is: Then, we arrange these 4 letters, considering that one letter is repeated twice. The number of distinct arrangements is: Thus, the total number of words in this case is: Case 3: Two letters repeated twice. We can select two letters to be repeated from {A, R}. There are 2 ways to choose the two repeated letters. Then, we arrange these 4 letters, considering that each letter is repeated twice. The number of distinct arrangements is: Thus, the total number of words in this case is: Total number of four-letter words: Now, we add up the results from all the cases: Thus, the total number of four-letter words that can be formed using the letters of "BARRACK" is .

To solve this problem, we need to arrange 6 boys and 4 girls around a round table such that no group of 2 boys and 1 girl sits together.

We consider the round table arrangement for boys. In a round table, for people, the arrangement is to account for rotational symmetries. Here, we have 6 boys which can be arranged in ways around the round table.

Now, having fixed the boys, they form spaces between them. There are 6 spaces (after each of the 6 boys) to place the girls. Since no group of 2 boys and 1 girl can sit together, we must ensure girls are not sitting directly between the boys. We effectively need to arrange the 4 girls in these spaces such that no space must be left with a "2 boys and 1 girl together" configuration.

One of the methods is to select 4 out of 6 spaces, filling them with the girls. This coincides with arranging the 4 girls in the selected spaces.

The number of ways to arrange 4 girls in 4 positions is given by .

To generalize this well, if we prevent 2 boys and 1 girl from sitting together, we need to use a repetitive arrangement which spices choosing and placing the girls correctly in the sequence of spaces living isolated from a pair of boys.

The captured logic gives us combining both expressions:

The total arrangements ensuring no group of 2 boys and 1 girl sits together is given by the expression .

This corresponds to the correct answer: .