Q) In an exciting finish to an one-day cricket match between India and Pakistan, the Indians require 10 runs in the last 3 balls to win. If any one of the scores 0,1,2,3,4,5,6 can be made from a ball and no wides or no-balls are bowled then in how many different sequences can the batsmen make exactly 10 runs?

My attempt:

I found the coeffecient of x^10 in the expansion of ( 1 + x + x^2 .... x^6)( 1 + x + x^2 .... x^6)(x + x^2 ..... x^6), which comes out to be 33. But the answer is 21.

Q) 5 Indian and 5 American couples meet at a party and shake hands. If no wife shakes hands with her husband and no Indian wife shakes hands with a male, then find the number of handshakes that take place in the party.

I got the answer 115 by simple combinations and inclusion-exclusion but answer is 135.

Both are AD Gupta problems hence even though I am pretty sure of my answers still wanna confirm.

1) if the q is to make 10 runs in 3 balls or less answer is 36..

if the q is to make 10 runs in exactly 3 balls(wat i mean is match ends only on last ball) then ans is 33..

i dunno where 21 came from!! i used logic..so in 2 diff methods ans comes 33..so i dun think it can be wrong

2)

well 1st consider indian females:

they make 5 shakes each..tht makes 25 shakes------(1)

american females:

totally 14 shakes/female: 5 wid Ind males + 4 wid american males(excluding husband) and 5 wid ind females(already incl above) so v r refering actually to 9 shakes/american female as the other 5 come in case 1...

so total num of shakes is 9*5=45..

not wat us left is shakes between indian males n american males which is 5*5=25

so total is 25+25+45=95...

where am i wrong???

oO I found my mistake. I wasnt counting the no. of handshakes of an American wife and the American husbands except her own, I just thought American wives dont shake hands with the American males. :\

So that adds 4*5 = 20 to the answer and I get 135 handshakes.

computer 001 , you forgot the cases where the Americans and Indians shake hands among themselves. So that adds 4* 5C2 to your answer.

oh shit!!!

yeah ok...neway wat abt the 1st Q??