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 $i$ , $1 \leq j \leq n - 1$

For $n \geq k, c (n, k)$ , where $c (n, k)$ has the same meaning as in question no.37, equals.

Options:

Answer:

Solution:

$C (n, k) = k (k - 1)^{n - 1} - c (n - 1, k)$

$= (k - 1)^{n} + (k - 1)^{n - 1} - C (n - 1, k)$ $C (n, k) - (k - 1)^{n} = (- 1) {C (n - 1, k) - (k - 1)^{n - 1}}$

$= (- 1)^{n - 3} {c (3, k) - (k - 1)^{3}}$

but $c (3, k) =$ number of mappings f in F for which $f (3) \neq = f (1)$

$∴ C (3, k) = k (k - 1) (k - 2)$

$∴ C (n, k) - (k - 1)^{n} = (- 1)^{n - 1} (k - 1) {k (k - 2) - (k - 1)^{2}}$

$(- 1)^{n} (k - 1)$

$∴ c (n, k) = (k - 1)^{n} + (- 1)^{n} (k - 1)$

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

⏱ 2mℹ️ Source: QB

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