fa892fa4-b511-4a63-977b-dfb9ec685e51 12 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

X n N M h M X 21 mat N M - n X - 21 mat * N n 21 mat / ≡

M X 21 mat N M - n X - 21 mat * N n 21 mat / M X combin N M - n X - combin * N n combin / ≡

Binomial_notation n k 21 mat ≡

n k 21 mat n! k! (n k - (! * / ≡

n! k! (n k - (! (* / n k combin ≡

x Γ x Gamma :

n k combin n 1 + Γ k 1 + Γ n k - 1 + Γ * / :

A deck of cards contains 20 cards: 6 red cards and 14 black cards. 5 cards are drawn randomly without replacement. What is the probability that exactly 4 red cards are drawn? The probability of choosing exactly 4 red cards is:

M 6 : X 4 : 21 sys

N 20 : n 5 : 21 sys

N M - n X - 21 mat 14121 mat

M X 21 mat N M - n X - 21 mat * N n 21 mat /

6421 mat 14121 mat * 20521 mat /

M X combin N M - n X - combin * N n combin / 0.0135

Combinations and permutations are similar since they are both subsets of items. However, internal order is significant in a permutation but not in a combination. combin(n,k) Returns the number of subsets each of size 'k' that can be formed from 'n' objects, where 'n' is greater or equal to 'k'. Each subset is known as a combination. 'n & k' are integers ... n & k ≥ 0

http://stackoverflow.com/questions/32493769/sets-of-all-disjoint-pairs

V e1 e2 Swap s V e1 el : V e1 el V e2 el : V e2 el s : 31 line :

V start pairs out MakePairs changed for Pairs ≡

V start pairs out Pairs Davide Carpi: 18 March, 2016 start V length > out out pairs 11 mat stack : sp1 start 1 + : sp2 start 2 + : j sp1 : j V length ≤ j j 1 + : V sp1 j Swap V sp2 pairs V start el V sp1 el 12 mat 11 mat augment out Pairs V sp1 j Swap 31 line for 31 line if completed 31 line :

V start el V sp1 el 12 mat 11 mat Nested pairwise ≡ V start el V sp1 el 12 mat suite over 'V' ≡ 21 line

v 16 range : 12345661 mat

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

Transit 'out' to Unest out 01 matrix :

pairs 10 matrix :

v 1 pairs out Pairs completed

out rows 15

out 8 el 1412 mat 3512 mat 2612 mat 13 mat

out 1212 mat 3412 mat 5612 mat 13 mat 1212 mat 3512 mat 4612 mat 13 mat 1212 mat 3612 mat 5412 mat 13 mat 1312 mat 2412 mat 5612 mat 13 mat 1312 mat 2512 mat 4612 mat 13 mat 1312 mat 2612 mat 5412 mat 13 mat 1412 mat 3212 mat 5612 mat 13 mat 1412 mat 3512 mat 2612 mat 13 mat 1412 mat 3612 mat 5212 mat 13 mat 1512 mat 3412 mat 2612 mat 13 mat 1512 mat 3212 mat 4612 mat 13 mat 1512 mat 3612 mat 2412 mat 13 mat 1612 mat 3412 mat 5212 mat 13 mat 1612 mat 3512 mat 4212 mat 13 mat 1612 mat 3212 mat 5412 mat 13 mat 151 mat

V 1212 mat 3412 mat 5612 mat 13 mat 1212 mat 3512 mat 4612 mat 13 mat 1212 mat 3612 mat 5412 mat 13 mat 1312 mat 2412 mat 5612 mat 13 mat 1312 mat 2512 mat 4612 mat 13 mat 1312 mat 2612 mat 5412 mat 13 mat 1412 mat 3212 mat 5612 mat 13 mat 1412 mat 3512 mat 2612 mat 13 mat 1412 mat 3612 mat 5212 mat 13 mat 1512 mat 3412 mat 2612 mat 13 mat 1512 mat 3212 mat 4612 mat 13 mat 1512 mat 3612 mat 2412 mat 13 mat 1612 mat 3412 mat 5212 mat 13 mat 1612 mat 3512 mat 4212 mat 13 mat 1612 mat 3212 mat 5412 mat 13 mat 151 mat :

V V Unest : a V 1 col Unest : b V 2 col Unest : c V 3 col Unest : 31 line U a b c augment : 31 line

V 1212 mat 3412 mat 5612 mat 1212 mat 3512 mat 4612 mat 1212 mat 3612 mat 5412 mat 1312 mat 2412 mat 5612 mat 1312 mat 2512 mat 4612 mat 1312 mat 2612 mat 5412 mat 1412 mat 3212 mat 5612 mat 1412 mat 3512 mat 2612 mat 1412 mat 3612 mat 5212 mat 1512 mat 3412 mat 2612 mat 1512 mat 3212 mat 4612 mat 1512 mat 3612 mat 2412 mat 1612 mat 3412 mat 5212 mat 1612 mat 3512 mat 4212 mat 1612 mat 3212 mat 5412 mat 153 mat

U 123456123546123654132456132546132654143256143526143652153426153246153624163452163542163254156 mat

Rework the program in convenient Smath code <= from unrecorded source N M Combin i 1 : k 1 : 12 mat i N M - 1 + (M^≤ j 1 M range tryV i j el i 1 - N M - 1 + (M j -^/ N M - 1 + mod trunc j + : for tryV i m el tryV i m 1 + el < (m 1 j 1 - product j 1 M range V k j el tryV i j el : for k k 1 + : 21 line computed if i i 1 + : 31 line while V 31 line :

Test 63 Combin :

Sanity check Test 123124125126134135136145146156234235236245246256345346356456203 mat

Observe the extreme suites: (4,2) => (1,2)... (3,4) (6,3) => (1,2,3) ... (4,5,6)

M NestCol a 0 : v M 1 col : ct 1 : j 1 M cols range a 1 ct el M j col : ct ct 1 + : 21 line for a 51 line :

M NestRow a 0 : v M 1 row : ct 1 : j 1 M rows range a ct 1 el M j row : ct ct 1 + : 21 line for a 51 line :

Test NestCol 11111111112222223334201 mat 22223334453334454455201 mat 34564565664565665666201 mat 13 mat

Test NestRow 12313 mat 12413 mat 12513 mat 12613 mat 13413 mat 13513 mat 13613 mat 14513 mat 14613 mat 15613 mat 23413 mat 23513 mat 23613 mat 24513 mat 24613 mat 25613 mat 34513 mat 34613 mat 35613 mat 45613 mat 201 mat

n k combin n! k! (n k - (! (* / :

combin(n,k) Returns the number of subsets each of size 'k' that can be formed from 'n' objects, where 'n' is greater or equal to 'k'. Each subset is known as a combination. 'n & k' are integers n & k ≥ 0

63 combin 20