Permutations And Combinations

Step 1: Identify the number of males and females available and the number to be selected. Available males = 5, Males to select = 2. Available females = 3, Females to select = 2.

Step 2: Calculate the number of ways to select the males. This is a combination problem since the order of selection does not matter. The number of ways to choose 2 males from 5 is given by the combination formula . Step 3: Calculate the number of ways to select the females. The number of ways to choose 2 females from 3 is: Step 4: Calculate the total number of ways to select the group. Since the selection of males and females are independent events, we multiply the number of ways for each selection (using the multiplication principle). Step 5: Compare the result with the given options. The calculated number of ways is 30, which matches option (B).

Step 1: Count the total number of letters in the word "DEPENDENT". The word has 9 letters: D, E, P, E, N, D, E, N, T.

Step 2: Count the frequency of each distinct letter. - D appears 2 times - E appears 3 times - P appears 1 time - N appears 2 times - T appears 1 time Total letters .

Step 3: Apply the formula for permutations with repetitions. The number of distinct permutations of objects where there are identical objects of type 1, identical objects of type 2, ..., identical objects of type k is given by: In this case, , and the counts of repeated letters are , , .

Step 4: Substitute the values into the formula.

Step 5: Calculate the factorial values and the final result.

Step 6: Compare the result with the given options. The calculated number of permutations is 15,120, which matches option (A).