If c ( n , k ) denote the number of functions in F satisfying f ( n )  = f ( 1 ) , then for n ≥ 4 , C ( n , k )

Mathematics - Permutation & Combination Question with Solution | TestHub

MathematicsPermutation & CombinationMiscellaneous/MixedHard2 minQB

MathematicsHardsingle choice

Passage / Comprehension

For a finite set $A$ , let $∣ A ∣$ denote the number of elements in the set $A$ . Also, let $F$ denote the set of all functions $f : {1, 2, \dots, n} \to {1, 2, \dots, k}$ where $n \geq 3$ and $k \geq 2$ , satisfying $f (i) \neq = f (i + 1)$ for every $i$ , $1 \leq j \leq n - 1$

If $c (n, k)$ denote the number of functions in F satisfying $f (n) \neq = f (1)$ , then for $n \geq 4, C (n, k)$

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Answer:

Solution:

Out of the total number of mappings in $F$ , the number of mappings which satisfy $f (n) = f (1)$ is the same as the number of mappings which satisfy $f (n - 1) \neq = f (1)$ . This number is $C (n - 1, k)$ .

Therefore,

Stream:JEESubject:MathematicsTopic:Permutation & CombinationSubtopic:Miscellaneous/Mixed

⏱ 2mℹ️ Source: QB

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