Looking for help with problem 5 Al has 9 different books He

Looking for help with problem 5?

Al has 9 different books. He decides to give 4 to Bob, 3 to Charles and 2 to Bob. In live many ways can this be done? In order to gain access to a private web sit, each person must have his or bet own password, where some digit is used 3 times and the remaining 3 digits are distinct. How many such passwords are there? A deck of 12 cards consists of 3 identical red cards, 3 identical blue cards, 3 identical grown cards and 3 identical yellow cards. The cards are mixed up (thrilled). Three cards are selected and the result is recorded. The three cards are then returned to the deck and the process is repeated. How many times must this be done to be guaranteed that there are three identical outcomes? how many 10-digit numbers contain five 1s, three 2s and two 3s? What is the coefficient of x in (2x - 1/2x)^T? Give the exact answer. Expand (a + b)^5, giving precise coefficients. What is the exact coefficient of a^4 b^3 c^2 in (a + b + c)^2?

Solution

We know that (a-x)ⁿ=aⁿ-C(n,1)a^n-1x+C(n,2)a^n-2x²+........................+(-1)^rC(n,r)a^n-rx^r+..........+(-1)ⁿxⁿ

The given equation (2x-1/2x)⁷

let we get coefficient of x from the term (-1)^rC(7,r)(2x)^7-r(1/2x)^r Here the degree of x is 1 therefore,

7-r-r=1 (from the term we calculate the degree of x)

-2r=-6 implies r=3

there for the coefficient for r=3 is (-1)³C(7,3)2⁴(1/2³⁾=-(7!/3!*4!)*2=-35*2=-70

therefore the coefficient of x is -70