A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. The number is:

Let the ten's digit be x.

Then, unit's digit = x + 2.

Number = 10x + (x + 2) = 11x + 2.

Sum of digits = x + (x + 2) = 2x + 2.

∴ (11x + 2)(2x + 2) = 144

⟹ 22x² + 26x - 140 = 0

⟹ 11x² + 13x - 70 = 0

⟹ (x - 2)(11x + 35) = 0

⟹ x = 2.

Hence, required number = 11x + 2 = 24.

Let the numbers be x and y.

Then, x + y = 25 and x - y = 13.

4xy = (x + y)² - (x- y)²

   = (25)² - (13)²

   = (625 - 169)

   = 456

∴ xy = 114.

We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).

∴ Required number of ways	= (7C3 x 6C2) + (7C4 x 6C1) + (7C5)
	= ❨ 7 x 6 x 5 x 6 x 5 ❩ + (7C3 x 6C1) + (7C2) 3 x 2 x 1 2 x 1
	= 525 + ❨ 7 x 6 x 5 x 6 ❩ + ❨ 7 x 6 ❩ 3 x 2 x 1 2 x 1
	= (525 + 210 + 21)
	= 756.

'LOGARITHMS' contains 10 different letters.

Required number of words	= Number of arrangements of 10 letters, taking 4 at a time.
	= 10P4
	= (10 x 9 x 8 x 7)
	= 5040.

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

∴ Number of ways of arranging these letters =	8!	= 10080.
	(2!)(2!)

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters =	4!	= 12.
	2!

∴ Required number of words = (10080 x 12) = 120960.

Let the ten's digit be x and unit's digit be y.

Then, number = 10x + y.

Number obtained by interchanging the digits = 10y + x.

∴ (10x + y) + (10y + x) = 11(x + y), which is divisible by 11.

Let the ten's and unit digit be x and	8	respectively.
	x

Then,	❨	10x +	8	❩	+ 18 = 10 x	8	+ x
			x			x

⟹ 10x² + 8 + 18x = 80 + x²

⟹ 9x² + 18x - 72 = 0

⟹ x² + 2x - 8 = 0

⟹ (x + 4)(x - 2) = 0

⟹ x = 2.

Let the numbers be x and y.

Then, xy = 9375 and	x	= 15.
	y

xy	=	9375
(x/y)		15

⟹ y² = 625.

⟹ y = 25.

⟹ x = 15y = (15 x 25) = 375.

∴ Sum of the numbers = x + y = 375 + 25 = 400.

Let the ten's digit be x and unit's digit be y.

Then, (10x + y) - (10y + x) = 36

⟹ 9(x - y) = 36

⟹ x - y = 4.

Let the ten's digit be x and unit's digit be y.

Then, x + y = 15 and x - y = 3   or   y - x = 3.

Solving x + y = 15   and   x - y = 3, we get: x = 9, y = 6.

Solving x + y = 15   and   y - x = 3, we get: x = 6, y = 9.

So, the number is either 96 or 69.

Hence, the number cannot be determined.