If you simply want to count the number of different ways to arrive at a different score, generating functions are a handy way of handling the data.

Take a power series for each method of scoring points. Let d = drop goal, t = try, and c = conversion; these score 3, 5, and 2 points, respectively. A conversion cannot be scored without a try, so let me amend that by letting c stand for a try with conversion, for 7 points, while t stands for try without conversion, for 5 points. Our respective power series are then:

1 + d[sup]3[/sup] + d[sup]6[/sup] + d[sup]9[/sup] + d[sup]12[/sup] + d[sup]15[/sup] + …

1 + t[sup]5[/sup] + t[sup]10[/sup] + t[sup]15[/sup] + t[sup]20[/sup] + t[sup]25[/sup] + …

1 + c[sup]7[/sup] + c[sup]14[/sup] + c[sup]21[/sup] + c[sup]28[/sup] + c[sup]35[/sup] + …

I’m not at all familiar with rugby, so I won’t enter this part of the debate, but if you wish to also include a 3 point penalty § as being distinct from a 3 point drop goal, we can also include:

1 + p[sup]3[/sup] + p[sup]6[/sup] + p[sup]9[/sup] + p[sup]12[/sup] + p[sup]15[/sup] + …

Now, generating functions can be useful for many things. For example, let’s say we want to know how many different ways there are of scoring n points. Take all four of the above power series, change all the variables to “x”, and multiply them together:

1 + 2*x^3 + x^5 + 3*x^6 + x^7 + 2*x^8 + 4*x^9 + 3*x^10 + 3*x^11 + 6*x^12 + 5*x^13 + 5*x^14 + 9*x^15 + 7*x^16 + 8*x^17 + 12*x^18 + 10*2^19 + 12*x^20 + 16*x^21 + 14*x^22 + 16*x^23 + 21*x^24 + 19*x^25 + 21*x^26 + 27*x^27 + 25*x^28 + 27*x^29 + 34*x^30 + 32*x^31 + 34*x^32 + 42*x^33 + 40*x^34 + 43*x^35 + 51*x^36 + 49*x^37 + 53*x^38 + 61*x^39 + 60*x^40 + 64*x^41 + 73*x^42 + 72*x^43 + 76*x^44 + 87*x^45 + 85*x^46 + 90*x^47 + 102*x^48 + 100*x^49 + 106*x^50 + 118*x^51 + 117*x^52 + 123*x^53 + 136*x^54 + 136*x^55 + 142*x^56 + 156*x^57 + 156*x^58 + 163*x^59 + 178*x^60 + 178*x^61 + 186*x^62 + 202*x^63 + 202*x^64 + 211*x^65 + 228*x^66 + 228*x^67 + 238*x^68 + 256*x^69 + 257*x^70 + 267*x^71 + 286*x^72 + 288*x^73 + 298*x^74 + 319*x^75 + 321*x^76 + 332*x^77 + 354*x^78 + 356*x^79 + 369*x^80 + 391*x^81 + 394*x^82 + 408*x^83 + 431*x^84 + 435*x^85 + 449*x^86 + 474*x^87 + 478*x^88 + 493*x^89 + 520*x^90 + 524*x^91 + 540*x^92 + 568*x^93 + 573*x^94 + 590*x^95 + 619*x^96 + 625*x^97 + 643*x^98 + 673*x^99 + 680*x^100 + …

The exponent gives the number of points scored, and the corresponding coefficient gives the number of distinct ways of scoring that many points. For example, there is 1 way of scoring 7 points, 6 ways of scoring 12 points, 106 ways of scoring 50 points, and 680 ways of scoring 100 points.

On the other hand, if we drop the distinction between drop goals and penalties, we get:

1 + x^3 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + 2*x^12 + 2*x^13 + 2*x^14 + 3*x^15 + 2*x^16 + 3*x^17 + 3*x^18 + 3*x^19 + 4*x^20 + 4*x^21 + 4*x^22 + 4*x^23 + 5*x^24 + 5*x^25 + 5*x^26 + 6*x^27 + 6*x^28 + 6*x^29 + 7*x^30 + 7*x^31 + 7*x^32 + 8*x^33 + 8*x^34 + 9*x^35 + 9*x^36 + 9*x^37 + 10*x^38 + 10*x^39 + 11*x^40 + 11*x^41 + 12*x^42 + 12*x^43 + 12*x^44 + 14*x^45 + 13*x^46 + 14*x^47 + 15*x^48 + 15*x^49 + 16*x^50 + 16*x^51 + 17*x^52 + 17*x^53 + 18*x^54 + 19*x^55 + 19*x^56 + 20*x^57 + 20*x^58 + 21*x^59 + 22*x^60 + 22*x^61 + 23*x^62 + 24*x^63 + 24*x^64 + 25*x^65 + 26*x^66 + 26*x^67 + 27*x^68 + 28*x^69 + 29*x^70 + 29*x^71 + 30*x^72 + 31*x^73 + 31*x^74 + 33*x^75 + 33*x^76 + 34*x^77 + 35*x^78 + 35*x^79 + 37*x^80 + 37*x^81 + 38*x^82 + 39*x^83 + 40*x^84 + 41*x^85 + 41*x^86 + 43*x^87 + 43*x^88 + 44*x^89 + 46*x^90 + 46*x^91 + 47*x^92 + 48*x^93 + 49*x^94 + 50*x^95 + 51*x^96 + 52*x^97 + 53*x^98 + 54*x^99 + 55*x^100 + …