multinomials and practice by Harvard Fat Chance

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Multinomials

key questions

how to understand the kind of problems multinomials try to tackle?

Video links

Bilibili

my note videos p25-26

Problems

9 students, split into 3 rooms (4, 3, 2 person per room), how many ways?

More problems

Suppose

• Strange
• steps = 7
• options = 7 unique letters
• multiplication and count sequence

Experiment

• steps = 7 slots
• options = 4 unique letters
• constraints
  • free repetition = no easy sequence with repetition = no $4^7$
  • 3 Es, 2 Ss, 1C, 1H are fixed, to be spreaded into 7 slots
  • inside 3 Es, order does not matter, so count collection, from 7 slots choose 3 slots for Es

Solution

• 3 Es count = $(^7_3)$
• 2 Ss count = $(^4_2)$
• 1 C count = $(^2_1)$
• 1 H count = $(^1_1)$
• between 4 groups, the order matters, we need sequence and multiplication here
  • $(^7_3) \times (^4_2) \times (^2_1) \times (^1_1) = \frac{7!}{4!3!} \frac{4!}{2!2!} \frac{2!}{1!} \frac{1!}{1!} = \frac{7!}{3!2!1!1!}$

Practice

Experiment

• steps = 15 people
• options = 3 kinds of salads (must repetition, can be $3^{15}$, order matters)
• each kind has fixed number 5, not any number allowed (15 ham cheese possible, also for the same kind sandwich the order does not matter)
• count = $(^{15}_5) (^{10}_5) (^5_5)$