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There are several chocolate ice-creams, several vanilla ice-creams, several mango ice-creams and several butterscotch ice-creams displayed in the ice-cream parlour. What is the number of ways of selecting 10 ice-creams in total from them?
Option: 1
$326$

Option: 2
$286$

Option: 3
$10^{4}$

Option: 4
Cannot be determined

Answers (1)

best_answer

Note the following:

The formula for the combination for the selection of the $\mathrm{x}$ items from the $\mathrm{y}$ different items is $\mathrm{=^{y}C_{x}=\frac{y!}{x!\left ( y-x \right )!}}$
The number of ways of allocating the $\mathrm{n}$ identical items among $\mathrm{r}$ head count with zero, one or more items is $\mathrm{=^{n+r-1}C_r-1}$

As per the available data, the number of identical types of ice-creams is $=1+1+1+1=4$

Note that the following is evident from the provided data.

The number from which the restricted combination is to be made is $\mathrm{=n+r-1=10+4-1=13}$ .
The number with which the restricted combination is to be made is $\mathrm{=r-1=4-1=3}$

Therefore, the required restricted combination is

$=\mathrm{^{n+r-1}C_{r-1}}$

$=\mathrm{^{13}C_{3}}$

$=\frac{13!}{ 3!10!}$

$=286$

Posted by

Rishabh

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