COMBINATIONS (mod4) 1. Patrick has assignments in 5 subjects. He can only do two assignments. In how many ways can he do two assignments? 2. In how many ways can a group of 5 men and 3 women be made out of a total of 10 men and 6 women? 3. A box contains 6 red, 5 blue and 3 white balls. In how many ways can we select 3 balls such that a. They are of different colors? b. They are all red? c. Two are blue and one is white? d. Exactly 2 are blue? e. None is white? f. At least two are white? BINOMIAL COEFFICIENTS 1. Expand the (2𝑚 − 2𝑏)3 using binomial coefficient. 2. Find for the coefficient of a5b5 ; (a - 4b )10 3. Find the 5th term after expanding the expression (3x – 4y)15 PIGEONHOLE PRINCIPLE Show that in a group of 27 English words, there must be at least two that begin with the same letter.

1.Number of ways in which he can do 2 asignments $= ^5P_2=\dfrac{5!}{(5-2)!}=\dfrac{120}{6}=20$

2.Number of ways $= ^{10}C_5\times ^{6}C_3=252\times 20=5040$

3.No. of ways of selcting 3 balls 

 a)if they are of different colour $=^6C_1\times ^5C_1 \times ^3C_1=90$

 b)If they all are red $= ^6C_3=20$

 c)if two are blues and one is white $= ^5C_2\times ^3C_1=10\times 3=30$

 d)if exactly 2 are blue $= ^5C_2\times ^9C_1=10\times 9=90$

 e)if none is white $= ^{11}C_3=132$

 f)if at least two are whute $=^3C_2\times ^{11}C_1=3\times11=33$

Binomial coeffecients'

1. $(2m-2b)^3$

   $=^3C_0(2m)^0(2b)^3+^3C_1(2m)^1(2b)^2+^3C_2(2m)^2(2b)^1+^3C_3(2m)^3(2b)^0\\ =8b^3+24mb^2+24m^2b+8m^3$

2. Given expression is -

   $(a-10b)^{10}$

The general term is-

  $T_{r+1}=^{10}C_r(a)r(-10b)^{10-r}$

 at r=5, coefficient of $a^5b^5= ^{10}C_5\times (-10)^5=-25200000$

3. The given expression is-

    $(3x-4y)^{15}$

  The $5^{th}$ term is = $^{15}C_4 (3x)^4 (-4y)^{15-4}$

Peigenhole principle

In this case, there are 26 pigeonholes (one for each letter in the English alphabet) but 27 pigeons, so we can immediately see that there must be at least one bucket with more than one word in it. But a bucket with more than one word implies that there are at least two words that start with the same letter.