In how many ways a selection can be made of atleast one fruit of 5 bananas, 4 mangoes and 4 almonds
To find the number of ways a selection can be made of at least one fruit from the given options, we need to consider two scenarios:
Scenario 1: Selecting only one type of fruit.
Scenario 2: Selecting multiple types of fruits.
Scenario 1:
To find the number of ways of selecting only one type of fruit, we can sum up the number of ways of selecting each fruit individually.
For bananas, there are 5 options: {1 banana}, {2 bananas}, {3 bananas}, {4 bananas}, {5 bananas}. Therefore, there are 5 ways of selecting only bananas.
Similarly, for mangoes, there are 4 options: {1 mango}, {2 mangoes}, {3 mangoes}, {4 mangoes}. Hence, there are 4 ways of selecting only mangoes.
Lastly, for almonds, there are 4 options: {1 almond}, {2 almonds}, {3 almonds}, {4 almonds}. So, there are 4 ways of selecting only almonds.
Scenario 2:
To find the number of ways of selecting multiple types of fruits, we need to consider all possible combinations.
Since there are three types of fruits, we can select 2 or 3 types of fruits simultaneously.
For selecting 2 types of fruits, we have a total of 3C2 ways (combining three fruits: bananas, mangoes, and almonds). This can be calculated using the formula: nCr = n! / ((n-r)! * r!). Therefore, 3C2 = 3! / ((3-2)! * 2!) = 3.
For selecting all 3 types of fruits, we have a total of 3C3 ways. This can be calculated as 3C3 = 3! / ((3-3)! * 3!) = 1.
So, there are 3 ways of selecting two types of fruits and 1 way of selecting all three types of fruits.
Therefore, to get the final answer, we add the number of ways from both scenarios:
Number of ways = Scenario 1 + Scenario 2
= (5 + 4 + 4) + (3 + 1)
= 20 + 4
= 24
Hence, there are 24 ways to make a selection with at least one fruit, considering the given options.
To find the number of ways a selection can be made with at least one fruit, we can use the concept of combinations.
Step 1: Calculate the total number of ways to select fruits without any restrictions.
To calculate the total number of ways, we need to consider each fruit separately.
For bananas: There are 5 bananas, so we can select either 0, 1, 2, 3, 4, or 5 bananas. This gives us a total of 6 possibilities.
For mangoes: There are 4 mangoes, allowing us to select either 0, 1, 2, 3, or 4 mangoes. This gives us a total of 5 possibilities.
For almonds: There are 4 almonds, giving us the same possibilities as mangoes.
Total number of ways to select fruits without any restrictions: 6 x 5 x 5 = 150.
Step 2: Subtract the cases where no fruit is selected.
To find the selection with at least one fruit, we need to exclude the cases where no fruit is selected.
For the bananas, if we select 0 bananas, we have 1 possibility. Similarly, we have 1 possibility each for mangoes and almonds.
Number of ways to select fruits without selecting any fruit: 1 x 1 x 1 = 1.
Step 3: Calculate the final number of ways.
Final number of ways = Total number of ways without any restrictions - Number of ways to select without selecting any fruit
Final number of ways = 150 - 1 = 149.
Therefore, there are 149 different ways to make a selection of at least one fruit out of 5 bananas, 4 mangoes, and 4 almonds.