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Out of the 10 packets, each containing a dozen fruits, the fruit seller placed 20 fruits in the showcase and threw away 50 which were damaged. In how many ways can the fruit seller arrange the fruits in the showcase with a total capacity of 100 fruits?
Option: 1
$\frac{70!}{20!50!}$

Option: 2
$\frac{100!}{20!50!}$

Option: 3
$\frac{150!}{20!50}$

Option: 4
$\frac{50!}{20!30!}$

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 $\mathrm{=^{y}C_{x}=\frac{y!}{x!(y-x)!}}$
The restricted combination for the selection of the $\mathrm{r}$ items from the $\mathrm{n}$ different items with $\mathrm{k}$ particular things always included is $\mathrm{^{n-k}C_{r-k}}$

As per the available data, the number of the fruits which are already arranged or damaged is

$=50+20$

$=70$

Since there are in total 120 fruits, the following is evident.

The number from which the restricted combination is to be made is $\mathrm{=n-k=120-70=50}$ .
The number with which the restricted combination is to be made is $\mathrm{=r-k=100-70=30}$

Therefore, the required restricted combination is

$\mathrm{=^{n-k}C_{r-k}}$

$\mathrm{=^{50}C_{30}}$

$=\frac{50!}{20!30!}$

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manish

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