42 ) Coefficient of x12 in the expansion of ( x3 + x2 + x)6
Gives the ans : )
EX 2
1 ) Find the no. of ways in which 3 distinct no.s can be selected from the set { 3 , 32,......,3101} so that they form a GP.
2 ) Find the no. of ways in which 12 identical coins can be distributed in 6 diff purses , if not mORE THAN 3 & NOT LESS THAN 1 coin goes in each purse.
3 ) There are 2n guests in a dinner party. If Master and Mistress of the house have fixed seats opposite one another & that there r 2 guests who can't sit beside one another .
Show that no of ways in which the company can be placed is (2n - 2 )! (4n2 - 6n + 4)
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8 Answers
621 ) Find the no. of ways in which 3 distinct no.s can be selected from the set { 3 , 32,......,3101} so that they form a GP.
Hint: This question is same as the number of ways in which you can select 3 numbers from the set {1,2,...101} so that they are in AP...
(Take a middle term) and find the number of APs with that middle term....
Can you solve it now?
2 ) Find the no. of ways in which 12 identical coins can be distributed in 6 diff purses , if not mORE THAN 3 & NOT LESS THAN 1 coin goes in each purse.
Let the number of coins in each purse be x1, x2,... x6
xi>=1 and <=3
so replace xi = 1+yi so that yi>=0 and <=2
x1+x2.. +x6=12
y1+y2+.. y6=6
where yi is less than equal to 2...
can you solve it now?
462yes. it does.. what i was saying is that by this logic, you can simplify such questions to
coeff of x^6 in (1+x+x^2)6
or even simpler to constant term in (1/x+x+1)6
:)
41) Nishant's expln contd ..
There can be 49 middle terms from 1 to 50 i.e 2 -> can form 1 AP
3-> 2 AP
......
50 --> 49 AP s
Sum of APs that can be formed from 1 to 50 is (49 * 50)/2
Sum of APs that can be formed from 51 to 101 is (50 * 51) / 2
Adding both total APs that can be formed is 2500 ANS
62no the second part i am afraid is not correct uttara...
Sum of APs that can be formed from 51 to 101 is (50 * 51) / 2
What is the number of AP's with middle term 99?
1in question 1...cant we divide into 2 groups..n then do it/?
evry two odd numbers will form the end of a triad in a ap and every two even numbers will form an ap wid its mean..
is this logic good??
4@ Nishant : 99 -> 1 ap with 99 as middle term
98 - > 2
........... 51
Dont we get 1 + 2 + ...50 ap s ?
Plz correct me ! if i'm wrong
@Rohit : I din't get u Plz explain taking an example
62sorry uttara.. my mistake.. your answer is correct..
I overlooked 101 :(