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Prove that .^(n-1)C(3)+.^(n-1)C(4) gt .^...

Prove that `.^(n-1)C_(3)+.^(n-1)C_(4) gt .^(n)C_(3)` if `n gt 7`.

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We have, `.^(n-1)C_(3)+.^(n-1)C_(4) gt .^(n)C_(3) " "[because.^(n)C_(r)+.^(n)C_(r-1)=.^(n+1)C_(r)]`
`hArr .^(n)C_(4)gt.^(n)C_(3)`
`hArr (n!)/(4!(n-4)!)gt(n!)/(3!(n-3)!)`
`hArr (1)/(4(n-4)!)gt(1)/((n-3)(n-4)!)" "[becausem!=m(m-1)!]`
`hArr n-3 gt 4 hArr n gt7`
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ARIHANT MATHS-PERMUTATIONS AND COMBINATIONS -Exercise (Subjective Type Questions)
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  2. Solve the equation 3 ^(x+1)C(2)+ ^(2)P(2)x=4^(x)P(2),x in N.

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  3. Number of positive terms in the sequence xn=195/(4Pn)-(n+3p3)/(P(n+1))...

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  4. Prove that .^(n-1)C(3)+.^(n-1)C(4) gt .^(n)C(3) if n gt 7.

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  5. In how many ways can a mixed doubles game in tennis be arranged from 5...

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  6. In how many ways, we can choose two teams of mixed double for a tennis...

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  7. A family consists of a grandfather, 5 sons and daughters and 8 grand c...

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  8. A tea party is arranged for 16 persons along two sides of a long table...

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  9. Every man who has lived on earth has made a certain number of handshak...

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  10. A train is going from cambridge to london stops at nine intermediate s...

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  11. How many 3-digit numbers can be formed by using the digits 1 to 9 if n...

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  12. A boat is to be manned by eight men, of whom 2 can only row on bow sid...

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  13. In how any different ways can a set A of 3n elements be partitioned in...

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  14. A square of n units is divided into n^(2) squares each of area 1 sq un...

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  15. How many sets of 2 and 3 (different) numbers can be formed by using nu...

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  16. There are n straight lines in a plane in which no two are parallel and...

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  17. How many 5-digit telephone numbers can be constructed using the digits...

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