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EXERCISE 3.2 · Q36

Q.Find n, if: [equation involving n, 8!, 6!, 3! — a stacked-fraction expression whose exact layout could not be confidently reconstructed from the extracted text]

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Concept understanding — Factorial Notation

For a natural number nn, the factorial of nn, written n!n!, is the product of all natural numbers from 1 up to nn: n!=1×2×3×⋯×(n−1)×nn! = 1\times2\times3\times\cdots\times(n-1)\times n. It can equally be read in the reverse (and often more useful) order, n!=n×(n−1)×(n−2)×⋯×2×1n!=n\times(n-1)\times(n-2)\times\cdots\times2\times1, which is what makes properties like n!=n×(n−1)!n!=n\times(n-1)! so natural: the factorial of a number is that number times the factorial of the number just below it. By convention, 0!0! is defined to equal 11 (even though 0 is not itself a natural number), which keeps formulas like nC0=1{}^nC_0=1 and nP0=1{}^nP_0=1 consistent without needing a special-case exception. Factorial notation grows extremely fast (10!10! is already over 3.6 million), so most factorial computations in practice are done by CANCELLING a shared tail between a numerator and a denominator (writing the larger factorial as extra factors times the smaller one) rather than by expanding both fully — this is the technique behind simplifying expressions like n!(n−r)!\dfrac{n!}{(n-r)!} …

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