Each set X- contains 5 elements and each set Y- contains 2 elements and Ur -120 Xr - S - Ur -1 nYr- If each element of S belong to exactly 10 of the Xr -s and to eactly 4 of Yr- s- then find the value of n-

It is given that each set X contains 5 elements and #8746;r=120Xr=S. #8756;nS=20 #215;5=100 But, it is given that each element of S belong to exactly ...

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Byju's Answer

Standard XII

Mathematics

Definition of Relations

Each set X, c...

Question

Each set X, contains 5 elements and each set Y, contains 2 elements and $\cup_{r = 1}^{20} X_{r} = S = \cup_{r = 1}^{n} Y_{r}$ . If each element of S belong to exactly 10 of the X_r's and to eactly 4 of Y_r's, then find the value of n.

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Solution

It is given that each set X contains 5 elements and $\cup_{r = 1}^{20} X_{r} = S$ .

$∴ n (S) = 20 \times 5 = 100$

But, it is given that each element of S belong to exactly 10 of the X_r's.

∴ Number of distinct elements in S = $\frac{100}{10} = 10$ .....(1)

It is also given that each set Y contains 2 elements and $\cup_{r = 1}^{n} Y_{r} = S$ .

$∴ n (S) = n \times 2 = 2 n$

Also, each element of S belong to eactly 4 of Y_r's.

∴ Number of distinct elements in S = $\frac{2 n}{4}$ .....(2)

From (1) and (2), we have

$\frac{2 n}{4} = 10 \Rightarrow n = 20$

Hence, the value of n is 20.

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Similar questions

Q. Each set X_r contains 5 elements and each set Y_r contains 2 elements and

\cup_{r = 1}^{20} X_{r} = S = \cup_{r = 1}^{n} Y_{r}

. If each element of S belongs to exactly 10 of the

X_{r}^{' s}

and to exactly 4 of the

Y_{r}^{' s},

then n is
(a) 10
(b) 20
(c) 100
(d) 50

Each set X, contains 5 elements and each set Y, contains 2 elements and $⋃_{r = 2}^{20} X_{r} = S = ⋃_{r = 1}^{n} Y_{r}$ . If each element of S belongs to exactly 10 of the $X_{r}^{^{'} s}$ and to exactly 4 of $Y_{r}^{^{'} s}$ then find the value of n.

A_{1}, A_{2}, \dots, A_{30}

are

30

sets, each having

5

elements and

B_{1}, B_{2}, \dots, B_{n}

are

n

sets each with

3

elements. If

30 ⋃ i = 1 A_{i} = n ⋃ j = 1 B_{j} = S

and each element of

S

belongs to exactly

10

of the

A_{i}

's and exactly

9

of the

B_{j}

's, then the value of

n

Suppose $A_{1}, A_{2}, . . ., A_{30}$ are thirty sets each having 5 elements and $B_{1}, B_{2}, . . ., B_{n}$ are n sets each having 3 elements. Let each element of S belongs to exactly 10 of $A_{i}^{'}$ s and exactly 9 of $B_{i}^{'}$ s. Then, find the value of n.