Evaluate the expression
(50)−(51)+(52)−(53)+(54)−(55)
Recall that the binomial coefficient is denoted by (nr) and is defined by
Notice that these are precisely the entries in the fifth row of Pascal's Triangle.
(50)=5!0!(5−0)!=1(51)=5!1!(5−1)!=5(52)=5!2!(5−2)!=10(53)=5!3!(5−3)!=10(54)=5!4!(5−4)!=5(55)=5!5!(5−5)!=1
Thus,
(50)−(51)+(52)−(53)+(54)−(55)=1−5+10−10+5−1=0
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