From 4 gentleman and 4 ladies a committee of 5 is to be formed which consist of a president,vice-president and three secretaries.
What will be the number of ways of selecting the committee with a maximum of 2 women and having at the maximum one woman holding one of the two posts( president,vice-president ) in the committee.
CASE 1: one lady &4 gent : 4c1 —> selecting one lady
4c4 —-> selecting 4 gents out of four
Arrangement : Presid Vp S S S —> no of ways :
Dividing by 3! as rearrangement btwn secretary position is useless
( 4c1 * 4c4 * 5! ) / 3! = 480/6= 80
CASE 2
Two lady & 3 gent (one lady will be in either vp or p )
: In this case we have to make sure that ATMOST one lady can sit in any one of two posts (vp , p)
4c1*2 —> Selecting one lady out of 4 then Lady 1 is selected she has either Vp or p position to select .
Then selecting one out of 4 gent so that it occupies remaining one of vp ,p posts —> 4c1 * 1c1
Rest two gents will have 2 post out of three positions And one lady will take the remaining post .
4 + 8 =12
CAse 3: Two lady and three gent ( lady is not sitting in vp or p post )
Selecting two gent to occupy p and Vp posts : 4c2 *2
Selecting two ladies out of 4 : 4c2
one gent is selected and placed in the reamaining position .
No of ways = 6 *2 * 6 = 72
72 + 12 + 80 = 162
Where am i wrong?
What will be the number of ways of selecting the committee with a maximum of 2 women and having at the maximum one woman holding one of the two posts( president,vice-president ) in the committee.
CASE 1: one lady &4 gent : 4c1 —> selecting one lady
4c4 —-> selecting 4 gents out of four
Arrangement : Presid Vp S S S —> no of ways :
Dividing by 3! as rearrangement btwn secretary position is useless
( 4c1 * 4c4 * 5! ) / 3! = 480/6= 80
CASE 2
Two lady & 3 gent (one lady will be in either vp or p )
: In this case we have to make sure that ATMOST one lady can sit in any one of two posts (vp , p)
4c1*2 —> Selecting one lady out of 4 then Lady 1 is selected she has either Vp or p position to select .
Then selecting one out of 4 gent so that it occupies remaining one of vp ,p posts —> 4c1 * 1c1
Rest two gents will have 2 post out of three positions And one lady will take the remaining post .
4 + 8 =12
CAse 3: Two lady and three gent ( lady is not sitting in vp or p post )
Selecting two gent to occupy p and Vp posts : 4c2 *2
Selecting two ladies out of 4 : 4c2
one gent is selected and placed in the reamaining position .
No of ways = 6 *2 * 6 = 72
72 + 12 + 80 = 162
Where am i wrong?