Combinations, Sets for Frequency System: Lotto 5/39, Lotto 6/49, Pick3, Pick4
By Ion Saliu, Founder of: Lottery Mathematics, Lotto Programming Science
Special lottery software: SkipDecaFreq, programs in the Bright / Ultimate application bundles.
Jackpot Lotto: Calculate Combinations from 3 Groups of Numbers
We divided the lotto numbers (jackpot games) in 3 groups of N1, N2, and N3 numbers. We work here with 2 lotto games that draw K numbers: K = 5 and K = 6. The lotto numbers are unique in each group: There is no overlap. We apply the combination formula to each group and finally multiply the 3 results.
C(N, M) is the general formula for combinations of N numbers taken M at a time.
N*(N1)*(N2)*(N3)* ... *(NM+1)
C(N, M) = 
1*2*3* ... *M
 Example: 5/39 lotto, with the following contents: Group #1 consists of 5 lotto numbers; group #2 has a total of 14 numbers; group #3 consists of 20 lotto numbers (total: 39 numbers). The frequency distribution is 1 – 1 – 3 (must total 5 as it is a 5number lotto game; must sum up to 6 for 6number lotto games):
 Group #1 combinations: C(N1, K1) = C(5, 1) = 5
 Group #2 combinations: C(N2, K2) = C(14, 1) = 14
 Group #3 combinations: C(N3, K3) = C(20, 3) = 1140
 Total combosnations (a favorite of mine!): 5 * 14 * 1140 = 79800.
The following tables show the amount of lottery combinations from 3 groups of numbers ordered by frequency. Lotto games analyzed: 5/39 and 6/49.
Lotto 5/39 Combinations for Frequency System
Group_1 = 5 #s

Group_2 = 14 #s

Group_3 = 20 #s

Total Combos

0

0

5

15504

0

5

0

2002

5

0

0

1

0

1

4

67830

0

4

1

20020

1

0

4

24225

1

4

0

5005

4

0

1

100

4

1

0

70

0

2

3

103740

0

3

2

69160

2

0

3

11400

2

3

0

3640

3

0

2

1900

3

2

0

910

1

1

3

79800

1

3

1

36400

3

1

1

2800

1

2

2

86450

2

1

2

26600

2

2

1

18200

Total C(39, 5)

~

~

575757

Lotto 6/49 Combinations for Frequency System
Group_1 = 6 #s

Group_2 = 18 #s

Group_3 = 25 #s

Total Combos

0

0

6

177100

0

6

0

18564

6

0

0

1

0

1

5

956340

0

5

1

214200

1

0

5

318780

1

5

0

51408

5

0

1

150

5

1

0

108

0

2

4

1935450

0

4

2

918000

2

0

4

189750

2

4

0

45900

4

0

2

4500

4

2

0

2295

1

1

4

1366200

1

4

1

459000

4

1

1

6750

0

3

3

1876800

3

0

3

46000

3

3

0

16320

1

2

3

2111400

1

3

2

1468800

2

1

3

621000

2

3

1

306000

3

1

2

108000

3

2

1

76500

2

2

2

688500

Total C(49, 6)

~

~

13983816

Pick Lotteries: Calculate Sets from 3 Groups of Digits
Although the odds are much lower, calculations in pick (digit, daily games) lotteries are more complex. The sets in these games have not only unique digits (as in jackpot lotto), but also repeat digits (doubles, triples, quadruples).
I. Calculating total unique sets is the easiest part. Total digits: N = 10; digits per set M = 3 (pick3) or M = 4 (pick4):
 Total unique M of N straight sets = (M! * C(N, M))
 Total single sets pick3 = 3! * 120 = 720
 Total single sets pick4 = 4! * 210 = 5040.
II. Calculating total duplicate sets is more complicated and involves more steps. The calculations are different from one type of duplication to another (doubles, triples, quadruples, quads).
 Pick3 doubles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with each of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple); thus, the second element of the calculation is 10 * 9 = 90. A 3digit set that contains a double and a single can be arranged in C(3, 2) = 3 ways; the final term of the calculation is 90 * 3 = 270 total doubles.
 Pick4 triples — All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with each of the 9 remaining digits (we do not combine a triple digit with itself as it would result in a quadruple); thus, the second element of the calculation is 10 * 9 = 90. A 4digit set that contains a triple and a single can be arranged in C(4, 3) = 4 ways; the final term of the calculation is 90 * 4 = 360 total triples.
 Pick4 1 double + 2 singles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with 2 of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple or a quadruple): 9 * 8 = 72; thus, the second element of the calculation is 10 * 72 = 720. A 4digit set that contains a double and 2 singles can be arranged in C(4, 2) = 6 ways; the final term of the calculation is 720 * 6 = 4320 total 1 double + 2 singles sets.
 Pick4 2 doubles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with doubles of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple or a quadruple); thus, the second element of the calculation is 10 * 9 = 90. A 4digit set that contains 2 distinct doubles can be arranged in C(41, 1) = 3 ways (e.g. 0011, 0101, 0110). The final term of the calculation is 90 * 3 = 270 total 2 doubles sets.
 Pick 3 triples — Each of the 10 digits forms a triple: 10 sets.
 Pick 4 quadruples — Each of the 10 digits forms a quad: 10 sets.
 Pick 3 Boxed: 120 (singles) + 90 (doubles) + 10 (triples) = 220 sets.
 Pick 4 Boxed: 210 (singles) + 720/2 (1 double + 2 singles) + 90/2 (2 doubles) + 90 (triples) + 10 (quads) = 715 sets.
Pick3 Sets for Frequency System
Group_1
2 digits

Group_2
3 digits

Group_3
5 digits

Single
Sets

Double
Sets

Triple
Sets

Total
Sets

0

0

3

60

60

5

125

0

3

0

6

18

3

27

3

0

0

0

6

2

8

0

1

2

180

45

0

225

0

2

1

90

45

0

135

1

0

2

120

30

0

150

1

2

0

36

18

0

54

2

0

1

30

30

0

60

2

1

0

18

18

0

36

1

1

1

180

0

0

180

Total Pick3

~

~

720

270

10

1000

Pick4 Sets for Frequency System
Group_1
2 digits

Group_2
3 digits

Group_3
5 digits

Single
Sets

1Pair
Sets

2Pair
Sets

Triple+1
Sets

Quad
Sets

Total
Sets

0

0

4

120

360

60

80

5

625

0

4

0

0

36

18

24

3

81

4

0

0

0

0

6

8

2

16

0

1

3

720

720

0

60

0

1500

0

3

1

120

360

0

60

0

540

1

0

3

480

480

0

40

0

1000

1

3

0

48

144

0

24

0

216

3

0

1

0

120

0

40

0

160

3

1

0

0

72

0

24

0

96

1

1

2

1440

360

0

0

0

1800

1

2

1

720

360

0

0

0

1080

2

1

1

360

360

0

0

0

720

0

2

2

720

540

90

0

0

1350

2

0

2

240

300

60

0

0

600

2

2

0

72

108

36

0

0

216

Total Pick4

~

~

5040

4320

270

360

10

10000

Pick 5 Lottery: Calculate ALL Sets in Quinto Lottery
Axiomatic one, I do not have a lot of software for pick 5 digit (Quinto) lottery. Therefore I will not divide the pick5 digits in groups based on frequency. I will calculate only the total sets: Singles, 1 double + 3 singles, 2 doubles + 1 single, 1 triple + 2 singles, 1 triple + 1 double, 1 quadruple + 1 single, quintets.
 Total unique M of N straight sets = (M! * C(N, M))
 Total single sets pick5 = 5! * 252 = 30240.
 1 double + 3 singles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with 3 of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple, a quadruple, or a quintet): 9 * 8 * 7 = 504; thus, the second element of the calculation is 10 * 504 = 5040. A 5digit set that contains a double and 3 singles can be arranged in C(5, 2) = 10 ways; the final term of the calculation is 5040 * 10 = 50400 total 1 double + 3 singles sets.
 2 doubles + 1 single (the most complicated case scenario) — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with each of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple, a quadruple, or a quintet); thus, the second element of the calculation is 10 * 9 = 90. The 2 doubles are combined with each of the 8 remaining digits. A 5digit set that contains 2 distinct doubles can be arranged in C(5, 2) = 10 ways; we also add to it the C(5, 1) = 5 of the single combinations: 10 + 5 = 15. The final term of the calculation is 90 * 8 * 15 = 10800 total 2 doubles + 1 single sets.
 1 triple + 2 singles — All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with 2 of the 9 remaining digits; thus, the second element of the calculation is 9 * 8 = 72; 72 * 10 = 720. A 5digit set that contains a triple and 2 singles can be arranged in C(5, 3) = 10 ways; the final term of the calculation is 720 * 10 = 7200 total 1 triple + 2 singles.
 1 triple + 1 double — All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with each of the 9 doubles; thus, the second element of the calculation is 10 * 9 = 90. A 5digit set that contains 1 triple and 1 double can be arranged in C(5, 3) = 10 ways; the final term of the calculation is 90 * 10 = 900 total 1 triple + 1 double.
 1 quadruplet + 1 single — All 10 digits can form a quad; therefore the first term is 10. Each of the 10 quadruples is combined with each of the 9 remaining digits; thus, the second element of the calculation is 10 * 9 = 90. A 5digit set that contains a quad and 1 single can be arranged in C(5, 4) = 5 ways; the final term of the calculation is 90 * 5 = 450 total 1 quadruplet + 1 single.
 Pick 5 quintets — Each of the 10 digits forms a quintuple: 10 sets.
 Grand total: 30240 + 50400 + 10800 + 7200 + 900 + 450 + 10 = 100,000 pick5 straight sets.
 Pick 5 Boxed: 252 (singles) + 840 (1 double + 3 singles) + 360 (2 doubles + 1 single) + 360 (1 triple + 2 singles) + 90 (1 triple + 1 double) + 90 (1 quad + 1 single) + 10 (quintets) = 2002 sets.
All Pick5 Lottery Sets: Distinct and Duplicate
Singles

1 Double +
3 Singles

2 Doubles +
1 Single

1 Triple +
2 Singles

1 Triple +
1 Double

1 Quad +
1 Single

Quintets

TOTAL

30240

50400

10800

7200

900

450

10

100000

Boxed Pick5 Lottery Sets
Singles

1 Double +
3 Singles

2 Doubles +
1 Single

1 Triple +
2 Singles

1 Triple +
1 Double

1 Quad +
1 Single

Quintets

TOTAL

252

840

360

360

90

90

10

2002

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