Question

Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives 3 blocks all of the same color is m/n, where m and n are relatively prime positive integers. What is m + n?

Answer 1

Given :- Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives 3 blocks all of the same color is m/n, where m and n are relatively prime positive integers. What is m + n ?

Solution :-

given that,

→ Total blocks each have = 5

So,

→ Total number of ways from which these blocks can be placed = (5!)^3

then,

→ Number of ways for which at least one block will get only one color will be = (5P1)(5C1)(4!)^3 - (5P2)(5C2)(3!)^3 + (5P3)(5C3)(2!)^3 - (5P4)(5C4) + (5P5)(5C5) = 306720

therefore,

→ m/n = 306720 /(120)^3 = 306720/1728000 = 71/400

hence,

→ m + n = 71 + 400 = 471 (Ans.)

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Answer 2

$\bf\green{\bigstar{QuestioN:-}}$

Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives 3 blocks all of the same color is m/n, where m and n are relatively prime positive integers. What  is  m + n?

$\bf\red{\bigstar{AnsweR:-}}$

Total no. of blocks each of them have = 5

So, no. of  the different ways it can be placed =  (5')³

Then , we can tell that

Number of ways in which atleast one colour block has been placed will be ,

$\bf\pink{:\implies{ (5A1)(5C1)(4')^3 - (5A2)(5C2)(3')^3 + (5A3)(5C3)(2')^3 - (5A4)(5C4) + (5A5)(5C5)}}$

$\bf\pink{:\implies306720}$

Then,

• $\purple\bf{ \dfrac{m}{n} = \dfrac{306720}{120^3}}$

• $\purple\bf{ \dfrac{m}{n} = \dfrac{306720}{1728000}}$

• $\purple\bf{ \dfrac{m}{n} = \dfrac{71}{400}}$

So,

• $\purple\bf{ m+n=71+400}$

• $\purple\bf{ m+n=471}$

Hence, we got the value of m + n.

The required answer = 471