some PnC plzzzzzzz puys help
1.in a certain test there are n question and atleast 2^(n-k) students give wrong answers to atleast k questions where k=1,2,3,4............,n now if total wrong answers are 2047 then then whats is the value of n ? options
10/11/12/13
2.no of ways in which a mixed double game can be arranged from amongst 9 couples such that no wife and husband play in same game ???options
756/1512/3024/none of these
3. given that n is odd the number of ways in which three numbers in AP can be selected in 1,2,3,4.............,n is
>(n-1)^2/4
>(n+1)^2/4
>(n+1)^2/2
>(n-1)^2/4
4.
(1) My take is 11
We have total number of wrong answers = 2^(n-n) + 2^(n-(n-1) + ...
= 2^0 + 2^1 + 2^2 + 2^3 + ..... + 2^(n-1)
= 2047
=> n = 12 as 2^0 + 2^1 + 2^2 + 2^3 + ...... + 2^10 = 2^11 - 1
(2) My take is 1512
Two men can be selected in 9C2 ways.
Now, we can select 2 women from 7 women as wives of those 2 men cannot be selected.
And then each man can make a team with any of two women. So, total two possibilities of games between 4 selected players.
So, total ways = 9C2 * 7C2 * 2 = 1512
(3) My take is (n-1)^2 / 4
We have numbers like 1, 2, 3, 4, ....., n
Now, we can AP with d = 1 as (1, 2, 3), (2, 3, 4), ....., (n-2, n-1, n)
=> (n-2) ways
With d = 2:
(1, 3, 5), (2, 4, 6), ....., (n-4, n-2, n)
=> (n-4) ways
With d = 3:
(1, 4, 7), (2, 5, 8 ), ........, (n-6, n-3, n)
=> (n-6) ways
Total = (n-2) + (n-4) + (n-6) + ..... + 1 ways
=> Sum of odd numbers from 1 till (n-2)
=> Sum of first (n-1)/2 odd number
=> ^2 ........ Sum of first 'x' odd numbers = x^2
=> (n-1)^2 / 4