A squad of 20 boys, which includes 2 sets of twins , is available for selection for a cricket team of 11 players. Calculate the number of different teams that can be selected if one set of twins is selected but neither twin from the other set is selected.
Well, it seems like we have some special rules for team selection here. Let's break it down step by step.
First, we need to select one set of twins, which means we have 2 options for that.
Next, we need to choose the remaining 9 players from the remaining 18 boys, which excludes the other set of twins. Since order doesn't matter, we can use combinations.
To calculate the number of combinations, we use the formula C(n, r) = n! / (r!(n-r)!), where n is the total number of options and r is the number of choices we need to make.
For our case, we have 18 boys to choose from and we need to choose 9 players, so we get C(18, 9) = 18! / (9!(18-9)!) = 48620.
Finally, we multiply the number of options for choosing one set of twins (2) with the number of options for choosing the remaining 9 players (48620). Therefore, the total number of different teams that can be selected is 2 * 48620 = 97240.
So, there are 97,240 different teams that can be selected under these conditions.
To calculate the number of different teams that can be selected, we can break down the problem into steps:
Step 1: Select one set of twins
Since there are 2 sets of twins available for selection, we have a choice of selecting either set. So, the number of choices for selecting one set of twins is 2.
Step 2: Select one twin from the selected set
Since we have selected one set of twins, we need to choose one twin from this set. Since there are 2 twins in a set, we have 2 choices for selecting one twin.
Step 3: Select the remaining players
After selecting one set of twins and one twin from that set, we need to select the remaining 9 players from the remaining 16 boys. The number of choices for this step is determined using combinations. We can calculate it as 16 choose 9 (written as "16C9").
The formula for combinations is: nCr = n! / (r! * (n-r)!), where n is the total number of objects and r is the number of objects to be chosen.
So, the number of choices for selecting the remaining players is:
16C9 = 16! / (9! * (16-9)!) = 16! / (9! * 7!) = (16 * 15 * 14 * 13 * 12 * 11 * 10) / (7 * 6 * 5 * 4 * 3 * 2 * 1) = 16 * 5 * 13 * 11 = 11,440.
Step 4: Multiply the choices from each step
To calculate the total number of different teams that can be selected, we multiply the number of choices from each step together:
Total number of teams = number of choices for step 1 * number of choices for step 2 * number of choices for step 3
= 2 * 2 * 11,440
= 45,760
Therefore, there are 45,760 different teams that can be selected if one set of twins is selected but neither twin from the other set is selected.
To calculate the number of different teams that can be selected, we need to account for the restrictions given in the question.
First, let's determine the number of ways to select the set of twins that will be included in the team. Since there are two sets of twins but only one set can be selected, we have 2 options.
Next, we need to select 9 players from the remaining 18 boys (excluding the set of twins and the other twin from the remaining set). This can be done in combination, denoted by "nCr". The formula for combination is:
nCr = n! / (r! * (n-r)!)
Where "n" represents the total number of options and "r" represents the number of selections.
In this case, we have 18 boys to choose from and we need to select 9, so the calculation is:
18C9 = 18! / (9! * (18-9)!)
Simplifying the equation:
18C9 = 18! / (9! * 9!)
To calculate factorials, we multiply a given number by all the positive integers less than it.
Using a calculator or a mathematical software, we can calculate 18! (18 factorial) as follows:
18! = 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Now, we can calculate 9! (9 factorial):
9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
To find the number of teams that can be selected, we divide the result of 18! by the product of 9! and (18-9)!, as follows:
18C9 = 18! / (9! * (18-9)!)
Now, let's calculate the number of teams:
18C9 = (18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / [(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) * (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)]
Simplifying the equation:
18C9 = (18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10) / (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)
Now, we can calculate the value of 18C9, which represents the number of different teams that can be selected. Using a calculator, we can perform the calculation:
18C9 = 48620
Therefore, there are 48,620 different teams that can be selected if one set of twins is chosen but neither twin from the other set is selected.