Consider the expression (x+5)^2
It expands out to x^2+10x+25 which has 3 terms.
The original expression has k = 2 terms inside the parenthesis and n = 2 is the exponent.
We then see that n+(k-1) = 2+(2-1) = 3 and k-1 = 2-1 = 1 is applied to this nCr combination formula to count the number of terms. You should find the result is 3.
If you kept k = 2 the same, and bumped things up to n = 3, then you should get 4 terms when you apply that modified nCr formula. That means something like (x+5)^3 will have 4 terms after simplifying everything and combining like terms.
For binomials of the form (A+B)^n, there are n+1 terms.
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This idea can be extended for k values that are larger than 2.
For problem 1, if n = 6 and k = 4, then we have
n+(k-1) = 6+(4-1) = 9
k-1 = 4-1 = 3
Then plugging those values into the formula you're given leads to 84. There are 84 terms in the expansion of (1+2x-7y+d)^6. Honestly, having this many terms is really difficult to verify by hand. It's not as easy compared to something like (x+5)^3. However, you can use CAS (computer algebra software) to expand out (1+2x-7y+d)^6 and confirm that there are indeed 84 terms. This is after all like terms are combined.
Problems 2 and 3 follow the same idea, but with different n and k values.
