A-teacher-conducts-a-test-for-five-students-He-provides-the-marking-scheme-and-asked-them-to-exchange-their-scripts-such-that-none-of-them-marks-his-own-script-How-many-ways-can-the-students-carry-o

Question Number 111394 by Aina Samuel Temidayo last updated on 03/Sep/20

A teacher conducts a test for five

students . He provides the marking

scheme and asked them to exchange

their scripts such that none of them

marks his own script . How many ways

can the students carry out the

marking ?

Commented by mr W last updated on 03/Sep/20

44

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

Solution please ?

Answered by mr W last updated on 03/Sep/20

t o t a l : 5!

a t l e a s t o n e s t u d e n t m a r k s h i s o w n s c r i p t :

C_{1}^{5} \times 4!

a t l e a s t t w o s t u d e n t s m a r k t h e i r o w n s c r i p t s :

C_{2}^{5} \times 3!

\dots

\Rightarrow 5! - C_{1}^{5} \times 4! + C_{2}^{5} \times 3! - C_{3}^{5} \times 2! + C_{4}^{5} \times 1! - 1

= 120 - 120 + 60 - 20 + 5 - 1

= 44

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

General formula for! n ?

Commented by mr W last updated on 03/Sep/20

o r

! 5 = 44

Commented by mr W last updated on 03/Sep/20

Commented by mr W last updated on 03/Sep/20

a s s h o w n a b o v e

! n = n! \underset{k = 0}{\sum^{n}} \frac{{(- 1)}^{k}}{k!}

w e h a v e

! n = (n - 1) [! (n - 1) +! (n - 2)]

! n = [\frac{n!}{e}]

Commented by mr W last updated on 03/Sep/20

w e s e e

\frac{! n}{n!} = \underset{k = 0}{\sum^{n}} \frac{{(- 1)}^{k}}{k!}

\lim_{n \to \infty} \frac{! n}{n!} = \underset{k = 0}{\sum^{\infty}} \frac{{(- 1)}^{k}}{k!} = \frac{1}{e}

Commented by Aina Samuel Temidayo last updated on 04/Sep/20

Thanks .