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Bilibili my note videos p21-24
Bilibili
my note videos p21-24
Experiment steps = 2 meat + 2 vegetables = 4 options = 7 meat + 6 vegetables = 13 how many unique pizzas can be counted? Solution divide the whole experiment into two parts: meat and vegetables meat part pizzas = collection count = $(^7_2)$ from 7 meat toppings select 2 as a unique pizza (remove order) vegetable part pizzas = collection count = $(^6_2)$ from 6 vegetable toppings select 2 as a unique pizza (remove order) each meat part can combine with different vegetable parts total collection count = meat part count x vegetable part count = $(^7_2) \times (^6_2)$ = 7x6/2x6x5/2 = 21x15=315 $(^7_4) \times (^6_0)$ = 7x6x5x4/(4x3x2x1) = 35 $(^7_3) \times (^6_1)$ = 7x6x5/(3x2x1) x 6 = 35x6 $(^7_1) \times (^6_3)$ = 7 x 6x5x4/(3x2x1) = 35x4 $(^7_0) \times (^6_4)$ = 1 x 6x5x4x3/(4x3x2x1) = 15 total = 315 + 350 + 35 + 15 = 715
Experiment steps = 5 students full options = 15 students constraint order is no important a chair has to be appointed Solution total count of 15 choose 5 collections = $(^{15}_5)$ upon which, how many ways to appoint a chair? still no worry about order for each collection, only pick one of the five student to be the chair = 5 choices target count = $(^{15}_5) \times 5$ = 15x14x13x12x11/(5x4x3x2x1) x 5 = 15x13x11x7 solution 2 first select 4 officials out of 15 students (ignore chair) = $(^{15}_4)$ second select chair out of 11 people = 11 total ways = $(^{15}_4) \times 11$ = 15x14x13x12 / (4x3x2x1) x 11 = 15x13x11x7
Experiment steps = 7 slots full options = 2, either E or N constraints four Es three Ns Solution total sequences = $2^7$ four Es and three Ns how many ways to get 4 slots out of 7 slots to be E $(^7_4)$ or $(^7_3)$ to focus on 3 Ns how many ways to get remaining 3 slots be N 1 as order between the 3 Ns is not important target count = 7x6x5x4/(4x3x2x1) = 35
How many ways to go from home to work? Experiment There are many paths each path can be seen as a sequence of repeated two letters (right or left; N or E) in fact to get from home to work, all ways have to have 4 Es and 3 Ns, and of course 7 steps in total Solution This is a hybrid problem = sequence (collection) sequence = order between N and E matters (outer part) collection = order inside Ns or Es does not matter (inner part) focus on 4 Es, worry no order = $(^7_4)$ remaining 3 slots for 3 Ns = $(^3_3 )$ = 1 target count = $(^7_4) \times (^3_3)$ = 35
How many paths out of the 35 above can pass through this shop in middle? experiment must passing through the shop = the first 3 steps must have 2 Es and 1 N part 1: between Es and Ns, we do sequence counting part 2: inside Es we do collection we do part 2 first from 3 steps we choose 2 for E = $(^3_2)$ we do part 1 = how many ways pass through shop count of Es x count of N = $(^3_2) \times (^1_1)$ = 3x2/(2x1) = 3 not yet finished, eventually arrive at work: how many ways from shop to work? count of Es x count of N = $(^4_2) \times (^2_2)$ = 4x3/(2x1)x1 = 6 join two parts before and after shop together 3 x 6 = 18
How many paths out of the 35 above can pass through this shop in middle?
15 students (7 boys, 8 girls), choose a committee with 4 students (2 boys, 2 girls), how many 15 students (7 boys, 8 girls), choose a committee with 4 students (not 4 boys, not 4 girls), how many same old question in the lecture above
15 students (7 boys, 8 girls), choose a committee with 4 students (2 boys, 2 girls), how many
15 students (7 boys, 8 girls), choose a committee with 4 students (not 4 boys, not 4 girls), how many
same old question in the lecture above
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