Find the number of ways the twelve students may be arranged in the exam hall.

Find the number of ways the students may be arranged if Helen and Nicky must sit so that one is directly behind the other (with no desk in between). For example Desk 5 and Desk 9.

Find the number of ways the students may be arranged if Helen and Nicky must not sit next to each other in the same row.

$12!{\text{ }}( = 479001600)$     A1

[1 mark]

METHOD 1

$8 \times 2 = 16$ ways of sitting Helen and Nicky, 10! ways of sitting everyone else     (A1)

$16 \times 10!$

$= 58060800$     A1

METHOD 2

$8 \times 1 \times 10!{\text{ }}( = 29030400)$ ways if Helen sits in the front or back row

$4 \times 2 \times 10!{\text{ }}( = 29030400)$ ways if Helen sits in the middle row     (A1)

Note:     Award A1 for one correct value.

$2 \times 29030400$

$= 58060800$     A1

[2 marks]

METHOD 1

$9 \times 2 \times 0!{\text{ }}( = 65318400)$ ways if Helen and Nicky sit next to each other     (A1)

attempt to subtract from total number of ways     (M1)

$12! - 9 \times 2 \times 10!$

$= 413683200$     A1

METHOD 2

$6 \times 10 \times 10!{\text{ }}( = 217728000)$ ways if Helen sits in column 1 or 4     (A1)

$6 \times 9 \times 10!{\text{ }}( = 195955200)$ ways if Helen sits in column 2 or 3     (A1)

$217728000 + 195955200$

$= 413683200$     A1

[3 marks]