Each card in a deck has an integer written on it, and the integers
on each of the 12 cards in the deck are consecutive. In a certain
game, the number points awarded for each turn is determined by
drawing two cards and multiplying the numbers shown on the
cards. If the points awarded in three turns are 40, 72, and 60,
all of the following could be the smallest numbered card in the
deck EXCEPT:
(A) -1
(B) 0
(C) 4
(D) 5
(E) 6

Eight machines, each working at the same constant rate,
together can complete a certain job in 9 days. How many
additional machines, each working at the same constant rate,
will be needed to complete the job in 6 days?
(A) 1
(B) 3
(C) 4
(D) 6
(E) 12

sukh wrote:
A certain academic department consists of 3 senior professors
and 6 junior professors. How many di¤erent committees of 3
professors can be formed in which at least one member of the
committee is a senior professor? (Two groups are considered
di¤erent if at least one group member is di¤erent.)
(A) 168
(B) 127
(C) 66
(D) 64
(E) 36

sukh wrote:2nd one answer is 4 they said we have to subtract 9 from 12 .
3rd , the answer is correct but why those options are forbidden
(Two groups are considered different if at least one group member is different )
Why are we calculating for 3 seniors and not 2 and 1 seniors

sukh wrote:
Eight machines, each working at the same constant rate,
together can complete a certain job in 9 days. How many
additional machines, each working at the same constant rate,
will be needed to complete the job in 6 days?
(A) 1
(B) 3
(C) 4
(D) 6
(E) 12

Answer C