Let the set of all relations on the set such that is reflexive and symmetric, and contains exactly 10 elements, be denoted by . Then the number of elements in is ______. [2025]

Question

Let the set of all relations $R$ on the set ${a, b, c, d, e, f},$ such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $S$ . Then the number of elements in $S$ is ______. [2025]

Smart Achievers · Accepted Answer

(105)

For relation to be reflexive, all the diagonal elements must be taken.

Out of the remaining 30 elements, there are 15 pairs, and we need 2 pairs such that R contains exactly 10 elements and is both reflexive and symmetric.

Hence, Number of ways $= {}^{15}C_{2} = 105$

Solution

Q.
Let the set of all relations $R$ on the set ${a, b, c, d, e, f},$ such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $S$ . Then the number of elements in $S$ is ______. [2025]

Ans.
(105)

For relation to be reflexive, all the diagonal elements must be taken.

Out of the remaining 30 elements, there are 15 pairs, and we need 2 pairs such that R contains exactly 10 elements and is both reflexive and symmetric.

Hence, Number of ways $= {}^{15}C_{2} = 105$

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Solution

Q. Let the set of all relations R on the set {a,b,c,d,e,f}, such that R is reflexive and symmetric, and R contains exactly 10 elements, be denoted by S. Then the number of elements in S is ______. [2025]

Ans. (105) For relation to be reflexive, all the diagonal elements must be taken. Out of the remaining 30 elements, there are 15 pairs, and we need 2 pairs such that R contains exactly 10 elements and is both reflexive and symmetric. Hence, Number of ways =C215=105

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Q.
Let the set of all relations $R$ on the set ${a, b, c, d, e, f},$ such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $S$ . Then the number of elements in $S$ is ______. [2025]

Ans.
(105)

For relation to be reflexive, all the diagonal elements must be taken.

Out of the remaining 30 elements, there are 15 pairs, and we need 2 pairs such that R contains exactly 10 elements and is both reflexive and symmetric.

Hence, Number of ways $= {}^{15}C_{2} = 105$