Use generating function to prove the identity
To prove the identity [ \sum_{k=0}^{n} \binom{r}{k} \binom{s}{n-k} = \binom{r+s}{n}, ] we can use generating functions. Consider the generating __________ _______ ________ ______ _______.
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यदि पुनरावृत्ति संबंध का हल
है , तो
और f(n) के मान ज्ञात कीजिए।
Find the general form of the solution to a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are 4, -2 and 3 with multiplicities 2, 1 and 3, respectively. The relation also has a non-homogeneous part which is a linear combination of 3n and 4n.
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90 owned a car, 35 owned a computer, 40 owned a house, 32 owned a car and a house, 21 owned a house and a computer, 26 owned a car and a computer, 17 owned all the three facilities.
i. How many people owned neither of the three.
ii. How many people owned only a car?
iii. How many people owned only a computer?
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ii. In a triangle , if
, then
.
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Find the generating function of the recurrence with initial conditions
.