Combinations calculation for the New York Pick 10

The combination calculation for the New York Pick 10 indicates how many combinations exist in a series of n numbers.
How many pairs are formed with ten numbers?
How many groups of 3 numbers can be formed with 6 numbers?
How many groups of 4, 5 or 6 numbers are there in the New York Pick 10 lottery?
This table provides the answers.

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NUMBER6 Num7 Num8 Num9 Num10 Num
1021012045101
114623301655511
1292479249522066
131.7161.7161.287715286
143.0033.4323.0032.0021.001
155.0056.4356.4355.0053.003
168.00811.44012.87011.4408.008
1712.37619.44824.31024.31019.448
1818.56431.82443.75848.62043.758
1927.13250.38875.58292.37892.378
2038.76077.520125.970167.960184.756
2154.264116.280203.490293.930352.716
2274.613170.544319.770497.420646.646
23100.947245.157490.314817.1901.144.066
24134.596346.104735.4711.307.5041.961.256
25177.100480.7001.081.5752.042.9753.268.760
26230.230657.8001.562.2753.124.5505.311.735
27296.010888.0302.220.0754.686.8258.436.285
28376.7401.184.0403.108.1056.906.90013.123.110
29475.0201.560.7804.292.14510.015.00520.030.010
30593.7752.035.8005.852.92514.307.15030.045.015
31736.2812.629.5757.888.72520.160.07544.352.165
32906.1923.365.85610.518.30028.048.80064.512.240
331.107.5684.272.04813.884.15638.567.10092.561.040
341.344.9045.379.61618.156.20452.451.256131.128.140
351.623.1606.724.52023.535.82070.607.460183.579.396
361.947.7928.347.68030.260.34094.143.280254.186.856
372.324.78410.295.47238.608.020124.403.620348.330.136
382.760.68112.620.25648.903.492163.011.640472.733.756
393.262.62315.380.93761.523.748211.915.132635.745.396
403.838.38018.643.56076.904.685273.438.880847.660.528
414.496.38822.481.94095.548.245350.343.5651.121.099.408
425.245.78626.978.328118.030.185445.891.8101.471.442.973
436.096.45432.224.114145.008.513563.921.9951.917.334.783
447.059.05238.320.568177.232.627708.930.5082.481.256.778
458.145.06045.379.620215.553.195886.163.1353.190.187.286
469.366.81953.524.680260.932.8151.101.716.3304.076.350.421
4710.737.57362.891.499314.457.4951.362.649.1455.178.066.751
4812.271.51273.629.072377.348.9941.677.106.6406.540.715.896
4913.983.81685.900.584450.978.0662.054.455.6348.217.822.536
5015.890.70099.884.400536.878.6502.505.433.70010.272.278.170
5118.009.460115.775.100636.763.0503.042.312.35012.777.711.870
5220.358.520133.784.560752.538.1503.679.075.40015.820.024.220
5322.957.480154.143.080886.322.7104.431.613.55019.499.099.620
5425.827.165177.100.5601.040.465.7905.317.936.26023.930.713.170
5528.989.675202.927.7251.217.566.3506.358.402.05029.248.649.430
5632.468.436231.917.4001.420.494.0757.575.968.40035.607.051.480
5736.288.252264.385.8361.652.411.4758.996.462.47543.183.019.880
5840.475.358300.674.0881.916.797.31110.648.873.95052.179.482.355
5945.057.474341.149.4462.217.471.39912.565.671.26162.828.356.305
6050.063.860386.206.9202.558.620.84514.783.142.66075.394.027.566
6155.525.372436.270.7802.944.827.76517.341.763.50590.177.170.226
6261.474.519491.796.1523.381.098.54520.286.591.270107.518.933.731
6367.945.521553.270.6713.872.894.69723.667.689.815127.805.525.001
6474.974.368621.216.1924.426.165.36827.540.584.512151.473.214.816
6582.598.880696.190.5605.047.381.56031.966.749.880179.013.799.328
6690.858.768778.789.4405.743.572.12037.014.131.440210.980.549.208
6799.795.696869.648.2086.522.361.56042.757.703.560247.994.680.648
68109.453.344969.443.9047.392.009.76849.280.065.120290.752.384.208
69119.877.4721.078.897.2488.361.453.67256.672.074.888340.032.449.328
70131.115.9851.198.774.7209.440.350.92065.033.528.560396.704.524.216
71143.218.9991.329.890.70510.639.125.64074.473.879.480461.738.052.776
72156.238.9081.473.109.70411.969.016.34585.113.005.120536.211.932.256
73170.230.4521.629.348.61213.442.126.04997.082.021.465621.324.937.376
74185.250.7861.799.579.06415.071.474.661110.524.147.514718.406.958.841
75201.359.5501.984.829.85016.871.053.725125.595.622.175828.931.106.355
76218.618.9402.186.189.40018.855.883.575142.466.675.900954.526.728.530
77237.093.7802.404.808.34021.042.072.975161.322.559.4751.096.993.404.430
78256.851.5952.641.902.12023.446.881.315182.364.632.4501.258.315.963.905
79277.962.6852.898.753.71526.088.783.435205.811.513.7651.440.680.596.355
80300.500.2003.176.716.40028.987.537.150231.900.297.2001.646.492.110.120