Combinatorial Selection of Fruits: A Mathematical Analysis

Imagine a scenario where you have a bag containing 4 mangoes and 5 oranges. How can you select fruits from this bag so that you take at least one mango and one orange? This problem can be solved using principles of combinatorics and a concept known as complementary counting. Let’s explore the step-by-step solution and even generalize the problem for any number of fruits.

Step-by-Step Solution for 4 Mangoes and 5 Oranges

To determine the total number of ways to select at least one mango and one orange, we can use complementary counting. This involves calculating the total number of combinations and then subtracting the invalid combinations (those that do not include at least one of mango and one orange).

Total Selections

Each fruit can either be included in the selection or not. Therefore, for 4 mangoes and 5 oranges:

Total selections (4 1) (5 1) 5 * 6 30.

Excluding Invalid Selections

Now, let's consider the invalid selections:

Selecting No Mangoes

If we select no mangoes, then we can only choose from the 5 oranges:

Selections with no mangoes (1 1) (5 1) 1 * 6 6

Selecting No Oranges

If we select no oranges, then we can only choose from the 4 mangoes:

Selections with no oranges (4 1) (1 1) 5 * 1 5

Adding these invalid selections:

Invalid selections 6 5 11

Valid Selections

Finally, we subtract the invalid selections from the total selections to get the valid selections:

Valid selections 30 - 11 19

Therefore, the total number of ways to select at least one mango and one orange from the bag is 19.

Generalizing the Problem for Any Number of Fruits

Now, let’s generalize this for selecting ( n ) fruits, where ( a ) is the number of mangoes and ( b ) is the number of oranges, such that ( a b n ).

Let ( a x_1 ) and ( b y_1 ), then:

( x_1 y_1 n-2 )

0 ≤ ( x ) ≤ ( n - 2 ), 0 ≤ ( y ) ≤ ( n - 2 )

The total number of non-negative integer solutions of ( ab n ) is the same as the total number of non-negative integer solutions of ( xy n-2 ).

Use the binomial coefficient:

(Total number of valid selections) ( binom{n-1}{n-2} binom{8}{7} cdot binom{7}{6} cdots binom{2}{0} )

Note that each term simplifies to:

8 7 ... 1 36 ways

This is the shortest solution to this problem.

Geometric Progression Explanation: The explanation can also be provided using geometric progressions, but it would be overly complicated to write in detail here.

Related Keywords:

Combinatorial selection Invalid selections Geometric progression