for problem #3, i'd do something like this. chances are there's a more efficient way, but this is how i saw the solution:
.......evaluating the last two summations we obtain
........now let's do some manipulation on the first summation
.......i didn't change anything, +2 - 4 = -2
......now evaluate the second summation
........a bit more manipulation on this summation should do it
.........again, I changed nothing, +4 - 2 = +2
.........i factored out a 2 from the first two terms
and now we have the desired summation
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This is my final question for math forum which i hope you can answer!
1. On page 345 of the textbook, the diagonal pattern of Pascalís Triangle illustrates that C(7, 5) = C(4, 4) + C(5, 4) + C(6, 4). Prove this relationship using the following methods.
a)numerically by using factorials
b)by reasoning, using the meaning of combinations