## Exercise 0.5

Do this and answer the question which is larger, the binomial coefficient C(15,5) or C(14,6)?

Solution:

Strangely enough they are the same.

 =-2 =A1+1 =B1+1 =C1+1 =D1+1 =E1+1 =F1+1 =G1+1 =A1+1 1 =A2+1 =C2+B2 =D2+C2 =E2+D2 =F2+E2 =G2+F2 =H2+G2 =A3+1 =C3+B3 =D3+C3 =E3+D3 =F3+E3 =G3+F3 =H3+G3 =A4+1 =C4+B4 =D4+C4 =E4+D4 =F4+E4 =G4+F4 =H4+G4 =A5+1 =C5+B5 =D5+C5 =E5+D5 =F5+E5 =G5+F5 =H5+G5 =A6+1 =C6+B6 =D6+C6 =E6+D6 =F6+E6 =G6+F6 =H6+G6 =A7+1 =C7+B7 =D7+C7 =E7+D7 =F7+E7 =G7+F7 =H7+G7 =A8+1 =C8+B8 =D8+C8 =E8+D8 =F8+E8 =G8+F8 =H8+G8 =A9+1 =C9+B9 =D9+C9 =E9+D9 =F9+E9 =G9+F9 =H9+G9 =A10+1 =C10+B10 =D10+C10 =E10+D10 =F10+E10 =G10+F10 =H10+G10 =A11+1 =C11+B11 =D11+C11 =E11+D11 =F11+E11 =G11+F11 =H11+G11 =A12+1 =C12+B12 =D12+C12 =E12+D12 =F12+E12 =G12+F12 =H12+G12 =A13+1 =C13+B13 =D13+C13 =E13+D13 =F13+E13 =G13+F13 =H13+G13 =A14+1 =C14+B14 =D14+C14 =E14+D14 =F14+E14 =G14+F14 =H14+G14 =A15+1 =C15+B15 =D15+C15 =E15+D15 =F15+E15 =G15+F15 =H15+G15 =A16+1 =C16+B16 =D16+C16 =E16+D16 =F16+E16 =G16+F16 =H16+G16 -2 -1 0 1 2 3 4 5 6 7 -1 1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 2 1 2 1 0 0 0 0 0 3 1 3 3 1 0 0 0 0 4 1 4 6 4 1 0 0 0 5 1 5 10 10 5 1 0 0 6 1 6 15 20 15 6 1 0 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 9 1 9 36 84 126 126 84 36 10 1 10 45 120 210 252 210 120 11 1 11 55 165 330 462 462 330 12 1 12 66 220 495 792 924 792 13 1 13 78 286 715 1287 1716 1716 14 1 14 91 364 1001 2002 3003 3432 15 1 15 105 455 1365 3003 5005 6435 16 1 16 120 560 1820 4368 8008 11440 17 1 17 136 680 2380 6188 12376 19448 18 1 18 153 816 3060 8568 18564 31824 19 1 19 171 969 3876 11628 27132 50388