Concept:Use the property C(n,r)=C(n,n−r) to pair and cancel terms.Explanation:The given expression is: C(51,21)−C(51,22)+C(51,23)−C(51,24)+C(51,25)−C(51,26)+C(51,27)−C(51,28)+C(51,29)−C(51,30).Apply C(51,r)=C(51,51−r).Thus C(51,21)=C(51,30), C(51,22)=C(51,29), C(51,23)=C(51,28), C(51,24)=C(51,27), C(51,25)=C(51,26).Substitute: C(51,30)−C(51,29)+C(51,28)−C(51,27)+C(51,26)−C(51,26)+C(51,27)−C(51,28)+C(51,29)−C(51,30).Group: (C(51,30)−C(51,30))+(C(51,29)−C(51,29))+(C(51,28)−C(51,28))+(C(51,27)−C(51,27))+(C(51,26)−C(51,26)).Each pair cancels to 0, so the sum is 0.Now check option C: C(51,51)−C(51,0). Using C(51,51)=C(51,0), this equals C(51,0)−C(51,0)=0, same as the expression.Answer:The value is 0, which matches option C: C(51,51)−C(51,0).