Question

(a) How many ways are there to pick a sequence of two different letters of the alphabet that appear in the word TUBA? Words to watch for: The word different tells you that you may not repeat any letter, so (T, T is not an acceptable sequence. The word sequence tells you that the ordering is important here: U, B) and B, U) are not the same sequence. (b) How many ways are there to pick a sequence of two different letters of the alphabet that appear in the word BASSOON?

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Solution :-

(a) :-

Given word: TUBA

Here we need to find out the number of ways are there to pick a sequence of two different letters of the alphabet that appear in the word TUBA.

Now consider, the number of possible combinations of 2 letters of word TUBA of 4 letters is  2

Now consider, For every possible combinations, there are 2 ways to pick a sequence, i.e, take two letters T and U, so the possible sequences are ( T, U ) and ( U,T ).

Hence, the number of ways are there to pick a sequence of two different letters is

Number of ways = 2 2

= AC

=  4-4

= 432 2 1(2)! 2

= 2 *rac{4*3*2*1}{2*1*(2*1)}

= 2

= 2 * 6

= 12

herefore The number of ways are there to pick a sequence of two different letters is 12

(b) :-

Given word: BASSOON

Here we need to find out the number of ways are there to pick a sequence of two different letters of the alphabet that appear in the word BASSOON.

The word BASSOON composition of B ightarrow 1, A ightarrow 1, S ightarrow 2 , O ightarrow 2, N ightarrow 1, so that there are 5 different letters. they are B, A, S, O, N .

Now consider, the number of possible combinations of 2 letters of word BASSOON of 5 letters is  inom{5}{2}

Now consider, For every possible combinations, there are 2 ways to pick a sequence, i.e, take two letters B and A, so the possible sequences are ( B, A ) and ( A,B ).

Hence, the number of ways are there to pick a sequence of two different letters is

Number of ways = 2 *inom{5}{2}

= 2 *5_{C} _{2}

=  2 *rac{5!}{2!*(5-2)!}

= 2 *rac{5*4*3*2*1}{2*1*(3)!}

= 120 2 1 (3 2 1)

= 120 2 12

= 2 10

= 20

herefore The number of ways are there to pick a sequence of two different letters is 20.

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