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Compute the terms CZ, 63, C4 of a recursively dened relation Ck = Ck_1 + kck_2 + 1 ,:for all integers 162 2 .(1) where initial conditions are C0 = 1...

Please ,Solve the following questions   

 

 

Q1   

Prove by the Principle of Mathematical Induction that

1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! - 1 for all natural numbers n.


Q2    

a)   How many license plates can be made using either four digits followed by five uppercase English letters or six uppercase English letters followed by three digits?

 

b)   Seven women and nine men are on the faculty in the mathematics department. How many ways are there to select a committee of five members of the department if at most two women must be on the committee?


c)  How many permutations of the letters ABCDEF contain the string CE?


Q3


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Q4

4.   What is the solution of the recurrence relation an=a(n1)+4a(n2)+4a(n3)

With a0=8,a1=6,a2=26?

 

Q5


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Q6

Find the first 3 terms in the expansion of (3xy)6 .

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3. Compute the terms CZ, 63, C4 of a recursively defined relation
Ck = Ck_1 + kck_2 + 1 ,:for all integers 162 2 ....(1) where initial conditions are C0 = 1 and C1 = 2 ............ :(2)

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5. Let a1, a2, a3, . and b1, b2, b3, satisfy the recurrence relation that the kth term
equals 3 times the (k-llst term for all integers k2 2: ak = 3ak_1, and bk 2 3bk_1 . But
suppose that the initial conditions for the sequences are differential = 2 and bl = 1.
Find a) a2, a3, a4. and b)b2, b3, b4.

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