(D) The number of diagonals in a polygon with $n$ sides is given by the formula $\frac{n(n-3)}{2}$.Given that the number of diagonals is $45$,we have

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Statement-$1$ is false,Statement-$2$ is true.

(D) The number of diagonals in a polygon with $n$ sides is given by the formula $\frac{n(n-3)}{2}$.Given that the number of diagonals is $45$,we have $\frac{n(n-3)}{2} = 45$.$n(n-3) = 90$.$n^2 - 3n - 90 = 0$.$(n-12)(n+9) = 0$.Since $n$ must be positive,$n = 12$.Thus,Statement-$1$ is false because the number of sides is $12$,not $10$.Statement-$2$ states that out of $n$ points,$2$ points can be selected in $^nC_2$ ways,which is a standard combinatorial result.Therefore,Statement-$1$ is false and Statement-$2$ is true.

Class 11 Mathematics Permutation and Combination Geometrical problems

Medium

Statement-$1$: If a polygon has $45$ diagonals,then the number of sides is $10$. Statement-$2$: Out of $n$ non-collinear points,$2$ points can be selected in $^nC_2$ ways.

A
Statement-$1$ is true,Statement-$2$ is true. Statement-$2$ is a correct explanation for Statement-$1$.
B
Statement-$1$ is true,Statement-$2$ is true. Statement-$2$ is not a correct explanation for Statement-$1$.
C
Statement-$1$ is true,Statement-$2$ is false.
D
Statement-$1$ is false,Statement-$2$ is true.

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Statement-$1$: If a polygon has $45$ diagonals,then the number of sides is $10$. Statement-$2$: Out of $n$ non-collinear points,$2$ points can be selected in $^nC_2$ ways.

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$10$ distinct points are taken on a circle. Then using these points:
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Statement $II$: The number of chords that can be formed is $45$
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