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Using the formula for combinations:(n−1)CrnCr=(n−1)!r!(n−1−r)!⋅r!(n−r)!n!=(n−1)!(n−r)!n!(n−1−r)!This simplifies to:(n−r)n=69=23 Step 3: Cross-multiply to find a relationship between n and rCross-multiplying gives:3(n−r)=2nExpanding ...
6. Using the Formula for the Sum of Cubes: The sum of the first m cubes is given by: (m(m+1)2)2 For m=n, we have: n−1∑r=0(r+1)3=(n(n+1)2)2 7. Substituting Back: Thus, we have: n−1∑r=0(r+1)3=(n(n+1)2)2 Therefore: 1(n+1)3⋅(n(n+1)2)2=...
Using the formula for combinations:(nr)=n!r!(n−r)!we can write:(n−1r)(nr)=(n−1)!r!(n−1−r)!n!r!(n−r)!=(n−1)!⋅(n−r)!n!⋅(n−1−r)!This simplifies to:(n−1)!n⋅(n−1−r)!⋅(n−r)!=(n−r)nThus, we have:n−rn=23 ...