please answer completely clearly and correctly thanks a)How many 16 bit strings

Question

please answer completely clearly and correctly thanks

a)How many 16 bit strings of 0's and 1's contain exactly 7 1's?

b) How many 16 bit strings of 0's and 1's contain at least one 1?

2)
a) How many ways can 2 integers from 1,2,...,100 be selected
so that their sum is even?

b) How many ways can 2 integers from 1,2,...,100 be selected
so that their sum is odd?

3) Calculate the number of integer solutions for the equation

    x1 + x2 + x3 = 20 , each xi ? integer; xi ? 0;

For ex. one solution is (x1, x2, x3) = (5,7,8)

4) How many integers from 1 to 999 have the sum of their
digits equal to 9?

For ex. the sum of digits in 54, 621, ... are equal to 9.

5) Calculate the number of ways to place 6 identical books
into 9 boxes.

Explanation / Answer

1)

a) 16C7=11440

b) Strings with at least one 1 = total strings -strings with no 1's =2^16-1=65535

2)

a)odd nos between 1 and 100=50

even nos. between 1 and 100=50

odd+odd=even and even+odd=even

So

no. of possibilities =50C2+50C1*50C1=1225+2500=3725

b)odd+even=odd

So

no. of ways= 50C1*50C1=2500

3)x1,x2,x3>=0

(20+3-1)C(3-1)=22C2=231

4)the number can be of the form abc, where a,b,c>=0 and a+b+c=9

So no. of ways=(9+3-1)C(3-1)=11C2=55

5) x1+x2+x3+x4+x5+x6+x7+x8+x9=6 ; where xi is the no. of books in i'th box

x1,x2,....,x9>=0

So

no. of ways =(6+9-1)C(9-1)=14C8=3003

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