TAOCP 1.2.5: Permutations and Factorials
Section 1.2.5 exercises: 18/18 solved.
Section 1.2.5. Permutations and Factorials
Exercises from TAOCP Volume 1 Section 1.2.5: 18/18 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [**] | solved | 5m32s | |
| 2 | [**] | verified | 1m17s | |
| 3 | [**] | verified | 3m40s | |
| 5 | [**] | verified | 37s | |
| 7 | [**] | verified | 45s | |
| 8 | [**] | verified | 1m12s | |
| 9 | [**] | verified | 3m35s | |
| 11 | [**] | solved | 34m34s | |
| 13 | [**] | verified | 8m39s | |
| 15 | [**] | verified | 2m56s | |
| 16 | [**] | verified | 31s | |
| 17 | [**] | verified | 2m36s | |
| 18 | [**] | solved | 7m04s | |
| 19 | [**] | verified | 36s | |
| 20 | [**] | verified | 2m30s | |
| 21 | [**] | verified | 2m26s | |
| 23 | [**] | verified | 36s | |
| 25 | [**] | verified | 39s |
TAOCP 1.2.5 Exercise 1
A shuffle of a 52-card deck is a permutation of 52 distinct objects.
TAOCP 1.2.5 Exercise 2
By Eq.
TAOCP 1.2.5 Exercise 3
Since \log_{10}(1000!
TAOCP 1.2.5 Exercise 5
Applying the refined form of Stirling’s approximation with $n=8$ gives 8!
TAOCP 1.2.5 Exercise 7
We have the generalized termial function defined for all real numbers $x$ by $x? = \sum_{k=1}^x k = 1 + 2 + \cdots + x.$ For integer $x$, this is exactly the usual arithmetic series formula, and for n...
TAOCP 1.2.5 Exercise 8
Let L_m=\frac{m^n m!
TAOCP 1.2.5 Exercise 9
Using the extension of the factorial function, x!
TAOCP 1.2.5 Exercise 11
Let $\nu_2(m)$ denote the exponent of $2$ in the prime factorization of $m$.
TAOCP 1.2.5 Exercise 13
Let S=\{1,2,\ldots,p-1\}, where $p$ is prime.
TAOCP 1.2.5 Exercise 15
The $(i,j)$ entry of the matrix is $i\times j$.
TAOCP 1.2.5 Exercise 16
Consider the infinite sum in Eq.
TAOCP 1.2.5 Exercise 17
Let P_N=\prod_{n=1}^{N}\frac{(n+\alpha_1)\cdots(n+\alpha_k)} {(n+\beta_1)\cdots(n+\beta_k)}.
TAOCP 1.2.5 Exercise 18
From Euler's limit formula for the factorial function (Equation 13), n!
TAOCP 1.2.5 Exercise 19
From Eq.
TAOCP 1.2.5 Exercise 20
By Exercise 19, for $x>0$, \Gamma_m(x) = \int_0^m\left(1-\frac{t}{m}\right)^m t^{x-1}\,dt.
TAOCP 1.2.5 Exercise 21
Let F_n=\sum_{j=0}^n B_{n,j}\,D_u^j w, where