All parts except (d) please. Let Ln be the number of ordered lists (without repe

Question

All parts except (d) please.

Let Ln be the number of ordered lists (without repetition) that can be formed from a set of n elements. That is, Problem 1.2.11 in our textbook asks you to prove that Ln is "very nearly n!e." Prove that (It follows immediately that , so that Ln is "asymptotic to" n!e.) For each positive integer n, show that Use this to prove that (This is stronger than the result above.) For a real number x, let [x] denote the integer nearest to x. For what values of n is Ln (ximil to [n!e]? (Optional) Use a computer spreadsheet or computer algebra system to calculate Ln and nle for 1

Explanation / Answer

// First of all try to understand the example

ordered list of size 0 that can be formed from set of n elements = 1

ordered list of size 1 that can be formed from set of n elements = nc1 (n choose 1)

ordered list of size 2 that can be formed from set of n elements = nc2 (n choose 2)

.

.

.

so Ln = submition of(n)k

= (1+x)^n

= nc0 + nc1*x + nc2*x^2 + nc3*x^3 ...

where x tending to 1.

now lim n tending to infinite

by induction , let

(1+x)^n/ n!

y = (1+x)/1 * (1+x)/2 * .....(1+x)/n

where x tending to 1 and n tending to infinite

2^n < n! < n^n

now y = (1+x)^n / ((n!)^n)^(1/n) (if taken uper limit)

= (1+x/n)^n

= e

fx = (x+1)^(1/x)x tends to infinite is e.

put lower limit for inequality.

3. (1+x)^n = n!e

n should be infinite.

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