Mathematics: A shipment of 12 television sets content to defective sets in how many ways can a hotel purchase five of these ads and receive at least two of the defective sets

A shipment of 12 television sets content to defective

dbanks701

26.04.2021

Total number of ways a hotel can purchase 5 of 12 sets and receive at least 2 defective sets = 288

Step-by-step explanation:

P.S - The exact question is -

Given - A shipment of 12 television sets contains 3 defective sets.

To find - In how many ways can a hotel purchase 5 of these sets and receive at least 2 of the defective sets?

Proof -

We know that The combination of n distinct objects taken k at a time is

$^{n} C_{k} = \frac{n!}{k! (n-k)!}$

Now,

The number of ways hotel can receive 2 defective sets among 5 that can be purchased is

$^{9} C_{3} * ^{3}C_{2} = \frac{9!}{3! (9-3)!} \frac{3!}{2! (3-2)!}$

⇒ $^{9} C_{3} * ^{3}C_{2}$ = 84(3) = 252

Now,

The number of ways that hotel receives 3 defective sets among 5 that are purchased is

$^{9} C_{2} * ^{3}C_{3} = \frac{9!}{2! (9-2)!} \frac{3!}{3! (3-3)!}$

⇒ $^{9} C_{2} * ^{3}C_{3}$ = 36(1) = 36

Now,

Total number of ways a hotel can purchase 5 of 12 sets and receive at least 2 defective sets = 252 + 36 = 288

⇒Total number of ways a hotel can purchase 5 of 12 sets and receive at least 2 defective sets = 288

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