Question

rohit dreams he is in a shop with an infinite amount of marbles. He is allowed to select n marbles. There are marbles of k different colors. From each color there are also infinitely many marbles. Rohit wants to have at least one marble of each color, but still there are a lot of possibilities for his selection. In his effort to make a decision he wakes up. Now he asks you how many possibilities for his selection he would have had. Assume that marbles of equal color can't be distinguished, and the order of the marbles is irrelevant.

Answer 1

Answer:

ⁿ⁻¹Ck₋1

Step-by-step explanation:

Hi,

Given that Rohit is supposed to select n marbles .

Given that there are k distinct color and at least 1  should be

selected from each color.

It implies that n > k otherwise selecting at least one from each

color wouldn't be possible.

Let X₁ be the number of ball to be selected of color 1

Let X₂ be the number of ball to be selected of color 2

........

Let Xk be the number of ball to be selected of color k

So, we need o calculate possible ways of selecting n balls such

that X₁ + X₂ +...........+ Xk = n------(1)

And also, Xi ≥ 1 ∀ i ,since there should be at least 1 ball of each

color.

Let us chose a new variable Yi = Xi - 1

Since Xi≥1 ∀ i, we get Yi ≥ 0 ∀ i

And equation (1), would become

(Y₁ + 1) + (Y₂ + 1)........+(Yk + 1) = n

Y₁ + Y₂ + ....... + Yk + k = n

Y₁ + Y₂ + ....... + Yk  = n - k, Yi ≥ 0

The solutions for the above equation is same as number of

non negative integral solutions satisfying the above equation.

Number of non-negative integral solution for

X₁ + X₂ +...........+ Xr = n are given by (n + r - 1)C r-1

Here r = k and n = n - k

Hence, the total number of ways are ⁿ⁻¹Ck₋1.

Hope, it helps !