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# System prepended metadata

title: 'Lab 10/31, Counting Practice'

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# Lab 10/31, Counting Practice

## Problem 1: Take-away
a. $C \times C \times C$
b. $C \times (C -1) \times (C - 2)$
c. $C \times C \times (C - 1) \times (C - 2) \times (C - 2)$
d. This approach does not work in scenarios where order among the cards in hand do not matter, eg one hand of 2, 6, 4 and another of 6, 4, 2 being equal. By finding ALL possible combinations, this approach counts every form of identical hands separately, thus inaccurately reporting the total number of hands.

## Problem 2: Dress-down
a. $a + s + p + h$
b. $a \times s \times p \times h$
c.  $a \times f \times p + 2a\times (h-f) \times s \times p$
d. $2^{hp}$

## Problem 3: The Perfect Fit
$(S_x \cap S_y) + (S_y \cap S_z) + (S_x \cap S_z) - S_a$
Check: 
Say we have c = 100 employees.
Of the 100, x = 50 can program in racket, y = 30 can write an inductive proof, and z = 20 are less than 175cm. Lastly, a = 5 candidates fulfilled all 3 criteria.
Of x, 10 can also write an inductive proof $(S_x \cap S_y)$, and 5 are also less than 175 cm $(S_x \cap S_z)$.
Of y, 10 are less than 175 cm $(S_y \cap S_z)$.
Thus, we would compute our quialified candidates by doing (10 + 10 + 5) - 5, giving us 20.
