If nPr = 3024 and nCr = 126 then find $n$ and r.

Step 1 (State Permutations/Combinations Relation): The number of permutations $nPr$ and combinations $nCr$ are related by:

Step 2 (Solve for r!): We substitute the given values $nPr=3024$ and $nCr=126$:

Since $4!=4\times 3\times 2\times 1=24$, we have:

Step 3 (Solve for n): Now we use the permutation formula for $r=4$:

We look for 4 consecutive integers whose product is 3024:

• $8\times 7\times 6\times 5=1680$ (too small)

• $9\times 8\times 7\times 6=3024$ (exactly matches!)

This gives $n=9$.

Step 4 (Conclusion): The values are $n=9$ and $r=4$, written as n=9, r=4.