Distribution of Lollipops to Students

Problem Statement

• A kindergarten teacher has a total of 29 students in her class.
• She possesses the following lollipops to distribute:
  • 8 yellow lollipops
  • 7 green lollipops
  • 6 psychedelic lollipops
  • 5 blue lollipops
  • 3 grey lollipops
• The task is to distribute one lollipop to each of 29 students.

Total Number of Lollipops

• Calculate the total number of lollipops available:
  • Total lollipops = 8 (yellow) + 7 (green) + 6 (psychedelic) + 5 (blue) + 3 (grey) = 29 lollipops.

Distribution Problem Analysis

• Since each student is to receive one lollipop, and the total number of lollipops exactly equals the number of students, this is a combinatorial distribution problem.
• The configuration resembles that of distributing indistinguishable objects (lollipops of the same color) into distinguishable slots (the students).

Methodology

• We can use the Multinomial Coefficient to calculate the ways in which the lollipops can be distributed.
• The formula for the Multinomial Coefficient is given by: $\frac{n!}{k<em>1! imes k</em>2! imes k<em>3! imes imes k</em>r!}$ Where:
  • n is the total number of items to distribute (lollipops),
  • k_i is the count of each distinct object (color of lollipops).

Applying the Formula

• For our scenario:
  • Total lollipops, n = 29.
  • The counts of lollipops per color are the respective k_i values:
  • Yellow: 8
  • Green: 7
  • Psychedelic: 6
  • Blue: 5
  • Grey: 3
• Plugging in the values:
  $ext{Ways} = \frac{29!}{8! imes 7! imes 6! imes 5! imes 3!}$

Conclusion

• The teacher can distribute the lollipops to the students in $\frac{29!}{8! imes 7! imes 6! imes 5! imes 3!}$ different ways.
• This represents a combinatorial arrangement, accounting for the distribution of lollipops of various types among the students.